Proof Proof of lemma 2.1

Fix $v\in W^{2,\infty }_{\mathrm B}(\Omega ;\mathbb {R}^N)\setminus \{0\}$ and define

\[ \rho_{\infty}(t):=\max_{x\in \overline{\Omega}}g\big(tv(x),t\mathrm{D} v(x)\big), \quad t\geq 0. \]

It follows that $\rho _{\infty }(0)=0$ and $\rho _{\infty }$ is continuous on $[0,\, \infty )$. We will now show that $\rho _{\infty }$ is strictly increasing. We first show it is non-decreasing. For any $s>0$ and $(\eta,\,P) \in \mathbb {R}^N \times \mathbb {R}^{N\times n} \setminus \{(0,\,0)\}$, our assumption (1.4)(c) implies

\begin{align*} 0\, & <\,\frac{C_7g(s\eta, sP)}{s} \\ & \leq \, \partial_\eta g (s\eta, sP)\cdot \eta + \partial_P g (s\eta, sP):P \\ & = \, \partial_{(\eta,P)} g (s\eta, sP) : (\eta,P) \\ & = \, \frac{\mathrm{d}}{\mathrm{d} s}\big(g(s\eta, sP)\big), \end{align*}

thus $s\mapsto g(s\eta,\, sP)$ is increasing on $(0,\, \infty )$. Hence, for any $x\in \overline {\Omega }$ and $t>s\geq 0$ we have $g(tv(x),\, t \mathrm {D} v(x)) \geq g(sv(x),\, s\mathrm {D} v(x))$, which yields,

\[ \rho_{\infty}(s)=\max_{x\in \overline{\Omega}}g\big(sv(x),s\mathrm{D} v(x)\big)\, \leq \, \max_{x\in \overline{\Omega}}g\big(tv(x),t\mathrm{D} v(x)\big)=\rho_{\infty}(t). \]

We proceed to demonstrate that $t\mapsto \rho _{\infty }(t)$ is actually strictly monotonic over $(0,\,\infty )$. Fix $t_0>0$. By Danskin's theorem [Reference Danskin13], the derivative from the right $\rho '(t_0^+)$ exists, and is given by the formula

\[ \rho_{\infty}'(t_0^+) = \, \max_{x\in {\Omega}_{t_0}}\Big\{ \partial_{(\eta,P)} g (t_0v(x), t_0 \mathrm{D} v(x)) : \big(v(x),\mathrm{D} v(x)\big) \Big\}, \]

where

\[ \Omega_{t_0}\,:=\, \Big \{ \overline{x}\in \overline{\Omega}\ :\ \rho_{\infty}(t_0)=g \big(t_0v(\overline{x}), t_0 \mathrm{D} v(\overline{x}) \big)\Big \}. \]

Hence, by (1.4)(c) we estimate

\begin{align*} \rho_{\infty}'(t_0^+)& =\frac{1}{t_0}\max_{x\in {\Omega}_{t_0}}\Big\{ \partial_{(\eta,P)} g (t_0v(x), t_0 \mathrm{D} v(x)) : \big(t_0v(x),t_0\mathrm{D} v(x)\big) \Big\} \\ & \geq \frac{C_7}{t_0}\max_{x\in {\Omega}_{t_0}}g \big(t_0v(x), t_0\mathrm{D} v(x)\big)\\ & = \frac{C_7}{t_0} \rho_{\infty}(t_0)\\ & >0. \end{align*}

This implies that $\rho _{\infty }$ is strictly increasing on $(0,\,\infty )$. Next, recall that $g$ is coercive by assumption (1.4)(b), namely $g(s\eta,\, sP)\to \infty$ as $s\to \infty$, for fixed $(\eta,\,P)\neq (0,\,0)$. Thus, for any fixed point $\overline {x}\in \Omega$ with $(v(\overline {x}),\,\mathrm {D} v(\overline {x}))\ne (0,\,0)$, which exists because by assumption $v\not \equiv 0$, we have

\[ \lim_{t\to \infty}\rho_{\infty}(t)\geq\lim_{t \to \infty} g(tv(\overline{x}), t \mathrm{D} v(\overline{x}))=\infty. \]

Since $\rho _\infty (0)=0$ and $\rho _{\infty }(t) \to \infty$ as $t\to \infty$, by continuity and the intermediate value theorem, there exists a number $t_\infty >0$ such that $\rho _{\infty }(t_{\infty })=1$, that is

\[ \big\|g\big(t_{\infty}v, t_{\infty} \mathrm{D} v\big)\big\|_{L^{\infty}(\Omega)}=1. \]

If $\|g(v,\, \mathrm {D} v)\|_{L^{\infty }(\Omega )}=1$, then $t_{\infty }=1$, as a result of the strict monotonicity of $\rho _{\infty }$. Now let us fix $p\in (n/\alpha,\, \infty )$ and define the continuous function

\[ \rho_p(t) \, :=\, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} g(tv, t \mathrm{D} v)^p \, \mathrm{d} \mathcal{L}^n , \quad t\geq 0. \]

Since $g(0,\,0)=0$, it follows that $\rho _p(0)=0$ and that

\[ \rho_p(t) = \frac{1}{\mathcal{L}^n(\Omega)}\int_{\{(v,\mathrm{D} v)\ne (0,0)\}} g(tv, t\mathrm{D} v)^p \, \mathrm{d} \mathcal{L}^n. \]

By Morrey's theorem and our assumptions, we have that $v\in C^1(\overline {\Omega };\mathbb {R}^N)\setminus \{0\}$, therefore $\mathcal {L}^n( \{(v,\,\mathrm {D} v)\ne (0,\,0)\} )>0$. Consider the family of functions $\{g(tv,\, t\mathrm {D} v)^p \}_{t>0}$, defined on $\{(v,\,\mathrm {D} v)\ne (0,\,0)\}\subseteq \Omega$. By the monotonicity of $s\mapsto g(s\eta,\, sP)$ on $(0,\,\infty )$ for $(\eta,\,P)\neq (0,\,0)$, for $s< t$ we have

\[ \text{$g(sv,\, s\mathrm{D} v)^p \leq g(tv,\, t\mathrm{D} v)^p$, \ on $\{(v,\,\mathrm{D} v)\ne (0,\,0)\}$}. \]

Since $g(tv,\, t\mathrm {D} v)^p \to \infty$ pointwise on $\{(v,\,\mathrm {D} v)\ne (0,\,0)\}$ as $t \to \infty$, by the monotone convergence theorem, we infer that

\[ \int_{\{(v,\mathrm{D} v)\ne (0,0)\}} g(tv, t\mathrm{D} v)^p \, \mathrm{d} \mathcal{L}^n \longrightarrow \infty, \]

as $t\to \infty$. As a consequence, $\rho _p(t)\to \infty$ as $t\to \infty$. Since $\rho _p(0)=0$, by the intermediate value theorem there exists $t_p>0$ such that $\rho _p(t_p)=1$, namely

\[ \big\|g(t_pv, t_p\mathrm{D} v)\big\|_{L^p(\Omega)}=1. \]

For the sake of contradiction, suppose that $t_p \not \to t_{\infty }$, as $p\to \infty$. In this case, there exists a subsequence $(t_{p_j})_1^{\infty }\subseteq (n/\alpha,\, \infty )$ and $t_0\in [0,\, t_{\infty }) \cup (t_{\infty },\, \infty ]$ such that $t_{p_j}\to t_0$ as $j\to \infty$. Further, $(t_{p_j})_1^{\infty }$ can assumed to be either monotonically increasing or decreasing. We first prove that $t_0$ is finite. If $t_0=\infty$, then the sequence $(t_{p_j})_1^{\infty }$ can be selected to be monotonically increasing. Therefore, by arguing as before, $g(t_{p_j}v,\, t_{p_j} \mathrm {D} v)\nearrow \infty$ as $j\to \infty$, pointwise on $\{(v,\,\mathrm {D} v)\ne (0,\,0)\}$, and the monotone convergence theorem provides the contradiction

\[ 1 = \lim_{j\to \infty}{-\hspace{-10.5pt}\displaystyle\int_{\Omega}} g(t_{p_j}v, t_{p_j} \mathrm{D} v)^{p_j} \, \mathrm{d} \mathcal{L}^n = {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} \lim_{j\to \infty} g(t_{p_j}v, t_{p_j} \mathrm{D} v)^{p_j} \, \mathrm{d} \mathcal{L}^n = \infty. \]

Consequently, we have that $t_0\in [0,\, t_{\infty }) \cup (t_{\infty },\, \infty )$. Since $(t_{p_j}v,\, t_{p_j}\mathrm {D} v)\to (t_0 v,\, t_0 \mathrm {D} v)$ uniformly on $\overline {\Omega }$ as $j\to \infty$, we calculate

\begin{align*} 1 \, & =\, \big\|g(t_{p_j}v, t_{p_j}\mathrm{D} v)\big\|_{L^{p_j}(\Omega)} \\ & =\,\big\|g(t_0v, t_0\mathrm{D} v)\big\|_{L^{p_j}(\Omega)} + \mathrm o(1)_{j\to\infty} \\ & =\, \big\|g(t_0v, t_0\mathrm{D} v)\big\|_{L^{\infty}(\Omega)} + \mathrm o(1)_{j\to\infty} \\ & =\, \rho_{\infty}(t_0) + \mathrm o(1)_{j\to\infty}. \end{align*}

By passing to the limit as $j\to \infty$, we obtain a contradiction if $t_\infty \ne t_0$, because $\rho _{\infty }$ is a strictly increasing function and $\rho _{\infty }(t_\infty )=1$. In conclusion, $t_p\to t_{\infty }$ as $p\to \infty$.

Proof Proof of lemma 2.2

Let us fix $p\in (n/ \alpha,\, \infty )$ and $v_0 \in W^{2,\infty }_{\mathrm B}(\Omega ;\mathbb {R}^N)$ where $v_0\, \not \equiv \, 0$. By application of lemma 2.1, there exists $t_p>0$ such that $\|g(t_p v_0,\, t_p\mathrm {D} v_0)\|_{L^p(\Omega )}=1$ implying that $t_p v_0$ is indeed an element of the admissible class of (1.10). Hence, we deduce that the admissible class is non empty. Further, by assumption (1.3)(b), $f$ is (Morrey) $2$-quasiconvex. We now confirm that $f^p$ is also (Morrey) $2$-quasiconvex function, as a consequence of Jensen's inequality: for any fixed $X\in \smash {\mathbb {R}^{N \times n^2}_s}$ and any $\phi \in W^{2,\infty }_0(\Omega ;\mathbb {R}^N)$, we have

\[ f^p(X)\leq \bigg( \, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}}f(X+\mathrm{D}^2 \phi) \, \mathrm{d} \mathcal{L}^n \!\bigg)^{\!p}\leq \, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} f(X+\mathrm{D}^2 \phi)^p \, \mathrm{d} \mathcal{L}^n. \]

By assumption by assumption (1.3)(d), we have for some new $C_5(p),\,C_6(p)>0$ that

\[ f(X)^p \, \leq \, C_5(p)|X|^{\alpha p}+C_6(p), \]

for any $X\in \smash {\mathbb {R}^{N \times n^2}_s}$. Moreover, by [Reference Zhou33, Theorem 3.6] we have that the functional $v \mapsto \|f(\mathrm {D}^2v)\|_{L^{p}(\Omega )}$ is weakly lower semi-continuous on $W^{2,\alpha p}(\Omega ;\mathbb {R}^N)$ and therefore the same is true over the closed subspace $W^{2,\alpha p}_{\mathrm B}(\Omega ;\mathbb {R}^N)$. Let $(u_i)_{1}^{\infty }$ be a minimizing sequence for (1.10). As $f\geq 0$, it is clear that $\inf _{i\in \mathbb {N}} \|f(\mathrm {D}^2u_i)\|_{L^{p}(\Omega )}\geq 0$. Since the admissible class is non-empty, the infimum is finite. Additionally, by (1.3)(d), we have the bound

\begin{align*} \inf_{i\in \mathbb{N}} \|f(\mathrm{D}^2u_i)\|_{L^{p}(\Omega)} & \leq \big\|f\big(\mathrm{D}^2(t_pv_0)\big)\big\|_{L^{p}(\Omega)} \\ & \leq \big\|C_5\big|t_p\mathrm{D}^2 v_0\big|^{\alpha}+C_6\big\|_{L^{\infty}(\Omega)} \\ & \leq C_5(t_p)^\alpha \|\mathrm{D}^2v_0\|^\alpha_{L^{\infty}(\Omega)}+C_6 \\ & < \infty. \end{align*}

Now we show that the functional is coercive in $W^{2,\alpha p}_{\mathrm B}(\Omega ;\mathbb {R}^N)$, arguing separately for either case of boundary conditions. By assumption (1.3)(d) and the Poincaré inequality, for any $u\in W^{2,\alpha p}_{\mathrm C}(\Omega ;\mathbb {R}^N)$ (satisfying $|u|=|\mathrm {D} u|=0$ on $\partial \Omega$), we have

\[ \bigg( \, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} \big|f(\mathrm{D}^2u)+C_3 \big|^p \, \mathrm{d} \mathcal{L}^n\bigg)^{\!\frac{1}{p}} \geq C_4\bigg( \, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}}|\mathrm{D}^2 u|^{\alpha p} \, \mathrm{d} \mathcal{L}^n \bigg) ^{\!\frac{1}{p}} \geq C_4'\|u\|^{\alpha}_{W^{1,\alpha p}(\Omega)}, \]

for a new constant $C_4'=C_4(p)>0$. Hence, for any $u\in W^{2,\alpha p}_{\mathrm C}(\Omega ;\mathbb {R}^N)$,

(2.1)\begin{equation} \|f(\mathrm{D}^2 u)\|_{L^p(\Omega)} \, \geq \, C_4'\big(\|u\|_{W^{2,\alpha p}(\Omega)}\big)^{\alpha} -C_3. \end{equation}

The above estimate is also true when $u\in W^{2,\alpha p}_{\mathrm H}(\Omega ;\mathbb {R}^N)$, but since in this case we have only $|u|=0$ on $\partial \Omega$, it requires an additional justification. By the Poincaré-Wirtinger inequality involving averages, for any $u\in W^{2,\alpha p}_{\mathrm H}(\Omega ;\mathbb {R}^N)$ we have

\[ \left\|\mathrm{D} u- {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} \mathrm{D} u \, \mathrm{d} \mathcal{L}^n \right\|_{L^{\alpha p}(\Omega)} \,\leq \, C\|\mathrm{D}^2u\|_{L^{\alpha p}(\Omega)}, \]

where $C=C(\alpha,\,p,\,\Omega )>0$ is a constant. Since $|u|=0$ on $\partial \Omega$, by the Gauss-Green theorem we have

\[ \int_{\Omega} \mathrm{D} u \, \mathrm{d} \mathcal{L}^n= \int_{\partial \Omega} u\otimes \hat{n} \, \mathrm{d} \mathcal{H}^{n-1}=0, \]

where $\mathcal {H}^{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure. In conclusion,

\[ \big\|\mathrm{D} u \|_{L^{\alpha p}(\Omega)} \, \leq \, C \|\mathrm{D}^2u\|_{L^{\alpha p}(\Omega)}, \]

for any $u\in W^{2,\alpha p}_{\mathrm H}(\Omega ;\mathbb {R}^N)$. The above estimate together with the standard Poincaré inequality applied to $u$ itself allow to infer that (2.1) holds for any $u\in W^{2,\alpha p}_{\mathrm B}(\Omega ;\mathbb {R}^N)$ in both cases of boundary conditions. Returning to our minimizing sequence, by standard compactness results, exists $u_p \in W^{2,\alpha p}_{\mathrm H}(\Omega ;\mathbb {R}^N)$ such that $u_i \, -\!\!\!\!\!-\!\!\!\!\rightharpoonup u _p$ in $W^{2,\alpha p}_{\mathrm B}(\Omega ;\mathbb {R}^N)$, as $i\to \infty$ along a subsequence of indices. Additionally, by the Morrey estimate we have that $u_i\longrightarrow u_p$ in $C^1(\overline {\Omega }; \mathbb {R}^N)$ as $i\to \infty$, along perhaps a further subsequence. Since $u\mapsto \|g(u,\, \mathrm {D} u)\|_{L^p(\Omega )}$ is weakly continuous on $W^{2,\alpha p}_{\mathrm B}(\Omega ;\mathbb {R}^N)$, the admissible class is weakly closed in $W^{2,\alpha p}(\Omega ;\mathbb {R}^N)$ and hence we may pass to the limit in the constraint. By weak lower semicontinuity of the functional, it follows that a minimizer $u_p$ which satisfies (1.10) does indeed exist.

Proof Proof of lemma 2.3

The result follows by standard results on Lagrange multipliers in Banach spaces (see e.g. [Reference Zeidler32, p. 278]), by utilizing assumption (1.3)(d), which guarantees that the functional is Gateaux differentiable.

Proof Proof of lemma 2.4

We begin by showing that $L_p>0$, namely the infimum over the admissible class of the $p$-approximating minimization problem is strictly positive, owing to the constraint and our assumptions (1.3)-(1.4). Indeed, there is only one map $u\in W^{2,\alpha p}_{\mathrm B}(\Omega ;\mathbb {R}^N)$ for which $\|f(\mathrm {D}^2u)\|_{L^p(\Omega )}=0$, namely $u_0\equiv 0$, but this is not an element of the admissible class since $\|g(u_0,\, \mathrm {D} u_0)\|_{L^p(\Omega )}=0$. Now consider the Euler–Lagrange equations in lemma 2.3 and select $\phi :=u_p$, to obtain

\begin{align*} & \,{-\hspace{-10.5pt}\displaystyle\int_{\Omega}} f(\mathrm{D}^2 u_p)^{p-1} \partial f(\mathrm{D}^2u_p): \mathrm{D}^2 u_p \, \mathrm{d} \mathcal{L}^n \\ & \quad = \, \lambda_p \, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} g(u_p, \mathrm{D} u_p)^{p-1}\Big(\partial_\eta g (u_p, \mathrm{D} u_p) \cdot u_p + \partial_P g (u_p, \mathrm{D} u_p) : \mathrm{D} u_p \Big) \, \mathrm{d} {\mathcal{L}}^{n}. \end{align*}

As $f,\,g\geq 0$ we can manipulate the respective assumptions (1.3)(c) and (1.4)(c) to produce the following bounds:

\begin{align*} C_1{-\hspace{-10.5pt}\displaystyle\int}_{\!\!\!\Omega} f(\mathrm{D}^2 u_p)^{p}\, \mathrm{d} \mathcal{L}^n \,& \leq {-\hspace{-10.5pt}\displaystyle\int_{\Omega}}\, f(\mathrm{D}^2u_p)^{p-1} \partial f(\mathrm{D}^2u_p): \mathrm{D}^2 u_p \, \mathrm{d} \mathcal{L}^n \\ & \leq \, C_2{-\hspace{-10.5pt}\displaystyle\int_{\Omega}} f(\mathrm{D}^2 u_p)^{p}\, \mathrm{d} \mathcal{L}^n,\\ C_7 \, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} g(u_p, \mathrm{D} u_p)^{p}\, \mathrm{d} \mathcal{L}^n & \leq \, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} g(u_p, \mathrm{D} u_p)^{p-1} \Big(\partial_\eta g (u_p, \mathrm{D} u_p) \cdot u_p + \\ & \quad + \, \partial_P g (u_p, \mathrm{D} u_p) : \mathrm{D} u_p \Big) \, \mathrm{d} {\mathcal{L}}^{n}\\ & \leq \, C_8 \, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} g(u_p, \mathrm{D} u_p)^{p}\, \mathrm{d} \mathcal{L}^n. \end{align*}

The above two estimates, combined with the Euler–Lagrange equations, imply that $\lambda _p>0$. Hence, we may therefore define $\Lambda _p:=(\lambda _p)^{\frac {1}{p}}>0$. We will now obtain the upper and lower bounds. We determine the lower bound as follows:

\begin{align*} C_1(L_p)^p \, & =\, C_1 \, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} f(\mathrm{D}^2 u_p)^{p}\, \mathrm{d} \mathcal{L}^n \\ & \leq \, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} f^{p-1}(\mathrm{D}^2u_p) \partial f(\mathrm{D}^2u_p): \mathrm{D}^2 u_p \, \mathrm{d} \mathcal{L}^n \\ & = \, \lambda_p \, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} g(u_p, \mathrm{D} u_p)^{p-1}\Big(\partial_\eta g (u_p, \mathrm{D} u_p) \cdot \phi + \, \partial_P g (u_p, \mathrm{D} u_p) : \mathrm{D} u_p \Big) \, \, \mathrm{d} {\mathcal{L}}^{n} \\ & \leq \, \lambda_p C_8. \end{align*}

Hence,

\[ \bigg(\frac{C_1}{C_8}\bigg)^{\frac{1}{p}}L_p\leq (\lambda_p)^{\frac{1}{p}}=\Lambda_p. \]

The upper bound is determined analogously, by reversing the direction of the inequalities. Combining both bounds, we obtain the desired estimate.

Proof Proof of proposition 2.5

Fix $p>n/\alpha,\, q\leq p$ and a map $v_0\in W^{2,\infty }_{\mathrm B}(\Omega ;\mathbb {R}^N)\setminus \{0\}$. Then, by lemma 2.1 there exists $(t_p)_{p\in (n/\alpha, \infty ]} \subseteq (0,\, \infty )$ such that $t_p \to t_{\infty }$ as $p\to \infty$ and satisfying $\|g(t_p v_0,\, t_p \mathrm {D} v_0)\|_{L^p(\Omega )}=1$ for all $p\in (n/\alpha,\, \infty ]$. By Hölder's inequality and minimality, we have the following estimate

\begin{align*} \big\|f(\mathrm{D}^2u_p) \big\|_{L^q(\Omega)} \, & \leq \, \big\|f(\mathrm{D}^2u_p) \big\|_{L^p(\Omega)} \\ & \leq \, \big\|f(t_p\mathrm{D}^2v_0) \big\|_{L^p(\Omega)} \\ & \leq\, \big\| f(t_p\mathrm{D}^2v_0) \big\|_{L^{\infty}(\Omega)} \\ & \leq K + \big\| f(t_\infty\mathrm{D}^2v_0) \big\|_{L^{\infty}(\Omega)}\\ & < \, \infty, \end{align*}

for some $K>0$. By (1.3)(d), we have the bound $f^q(X)\geq C_4(q)|X|^{\alpha q}- C_3(q)$ for some constants $C_3(q),\, C_4(q)>0$ and all $X\in \mathbb {R}^{N \times n^2}_s.$ By the previous bound, we conclude that

\[ \sup_{q\geq p}\|\mathrm{D}^2u_p\|_{L^{\alpha q}(\Omega)}\leq C(q)< \infty, \]

for some $q$-dependent constant. By arguing as in the proof of lemma 2.2 through the use of Poincaré inequalities, we can conclude in both cases of boundary conditions with the bound

\[ \sup_{q\geq p}\|u_p\|_{W^{2, \alpha q}(\Omega)}\leq C(q)< \infty, \]

for a new $q$-dependent constant $C'(q)>0$. Standard compactness in Sobolev spaces and a diagonal sequence argument imply the existence of a mapping

\[ u_\infty \in \bigcap_{n/\alpha< p<\infty} W^{2, \alpha p}_{\mathrm B}(\Omega; \mathbb{R}^N) \]

and a subsequence $(p_j)_1^\infty$ such that the desired modes of convergence hold true as $p_j\to \infty$ along this subsequence of indices. Fix a map $v\in W^{2, \infty }_{\mathrm B}(\Omega ; \mathbb {R}^N)$ satisfying the required constraint, namely $\|g(v,\, \mathrm {D} v) \|_{L^{\infty }(\Omega )}=1$. In view of lemma 2.1, there exists $(t_p)_{p\in (n/\alpha, \infty )}\subseteq (0,\, \infty )$ satisfying that $t_p\to 1$ as $p\to \infty$, and additionally $\|g(t_p v,\, t_p \mathrm {D} v)\|_{L^p(\Omega )}=1$ for all $p>n/\alpha$. By Hölder's inequality, the definition of $L_p$ and minimality, we have

\[ \big\|f(\mathrm{D}^2 u_p)\big\|_{L^q(\Omega)} \, \leq \, \big\|f(\mathrm{D}^2 u_p)\big\|_{L^p(\Omega)} = \, L_p \, \leq \, \big\|f(t_p\mathrm{D}^2 v)\big\|_{L^p(\Omega)}, \]

for any such $v$. By the weak lower semi-continuity of the functional on $W^{2, \alpha q}_{\mathrm B}(\Omega ; \mathbb {R}^N)$, we may let $p_j \to \infty$ to obtain

\begin{align*} \big\|f(\mathrm{D}^2u_{\infty})\big\|_{L^q(\Omega)} \, & \leq \, \liminf_{p_j\to \infty}L_p \\ & \leq \, \limsup_{p_j\to \infty} L_p \\ & \leq \, \limsup_{p_j \to \infty} \|f(t_{p_j}\mathrm{D}^2 v)\|_{L^p(\Omega)} \\ & =\, \|f(\mathrm{D}^2 v)\|_{L^{\infty}(\Omega)}. \end{align*}

Now we may let $q\to \infty$ in the above bound, hence producing

\[ \big\|f(\mathrm{D}^2 u_{\infty})\big\|_{L^{\infty}(\Omega)} \, \leq \, \liminf_{p_j\to \infty}L_p \, \leq \, \limsup_{p_j\to \infty} L_p \, \leq \, \|f(\mathrm{D}^2 v)\|_{L^{\infty}(\Omega)}. \]

for all mappings $v\in W^{2, \infty }_{\mathrm B}(\Omega ; \mathbb {R}^N)$ satisfying the constraint $\|g(v,\, \mathrm {D} v)\|_{L^{\infty }(\Omega )}=1$. If we additionally show that in fact $u_{\infty }$ satisfies the constraint in (1.1), then the above estimate shows both that it is the desired minimizers (by choosing $v:=u_\infty$), and also that the sequence $(L_{p_j})_1^\infty$ converges to the infimum. Now we show that this is indeed the case. In view of assumption (1.3)(d), the previous estimate implies also that $\mathrm {D}^2u_{\infty }\in \smash {L^{\infty }(\Omega ; \mathbb {R}_s^{N \times n^2})}$, which together with Poincaré inequalities (as in the proof of lemma 2.2) shows that in fact $u_{\infty } \in W^{2, \infty }_{\mathrm B}(\Omega ; \mathbb {R}^N)$. By the continuity of the function $g$ and the fact that $u_{p} \longrightarrow u_\infty$ in $C^1 (\overline {\Omega };\mathbb {R}^N )$, we have

\begin{align*} 1& = \, \|g(u_p, \mathrm{D} u_p)\|_{L^p(\Omega)}\\ & = \, \|g(u_{\infty}, \mathrm{D} u_{\infty})\|_{L^p(\Omega)} + \|g(u_p, \mathrm{D} u_p)\|_{L^p(\Omega)}-\|g(u_{\infty}, \mathrm{D} u_{\infty})\|_{L^p(\Omega)}\\ & = \, \|g(u_{\infty}, \mathrm{D} u_{\infty})\|_{L^p(\Omega)} + \,\mathrm O\Big(\|g(u_p, \mathrm{D} u_p)- g(u_{\infty}, \mathrm{D} u_{\infty})\|_{L^{\infty}(\Omega)}\Big)\\ & \longrightarrow \|g(u_{\infty}, \mathrm{D} u_{\infty})\|_{L^{\infty}(\Omega)}, \end{align*}

as $p_j\to \infty$. Consequently, $u_{\infty }$ satisfies the constraint, and therefore lies in the admissible class of (1.1). Since $v$ was arbitrary in the energy bound, we conclude that $u_{\infty }$ in fact solves (1.1). let us now define

\[ \Lambda_{\infty}:=\big\|f(\mathrm{D}^2 u_{\infty})\big\|_{L^{\infty}(\Omega)}. \]

We now show that $\Lambda _{\infty }>0$. Indeed, by our assumptions (1.3)–(1.4), there is only one map in $W^{2,\infty }(\Omega ;\mathbb {R}^N)$ satisfying $\|f(\mathrm {D}^2u_0)\|_{L^{\infty }(\Omega )}=0$ and $|u_0|\equiv 0$ on $\partial \Omega$, namely the trivial map $u_0\equiv 0$, but $u_0$ is not contained in the admissible class of (1.1) because $\|g(u_0,\, \mathrm {D} u_0)\|_{L^{\infty }(\Omega )}=0$. We now show that $\Lambda _p \longrightarrow \Lambda _{\infty }$ as $p_j \to \infty$. By our earlier energy estimate, we have $L_p\longrightarrow \Lambda _{\infty }$ as $p_j \to \infty$. By lemma 2.4, we have

\[ 0< \, \lim_{p_j\to \infty} \bigg(\frac{C_1}{C_8}\bigg)^{\frac{1}{p}}L_p \, \leq \, \lim_{p_j\to \infty} \Lambda_p \, \leq \, \lim_{p_j\to \infty} \bigg(\frac{C_2}{C_7}\bigg)^{\frac{1}{p}}L_p, \]

and therefore $\Lambda _p \longrightarrow \Lambda _{\infty }$ as $p_j \to \infty$. To complete the proof we must deduce the claimed bounds for $\Lambda _{\infty }$. We first establish the lower bound. By utilizing the Poincaré and Poincaré-Wirtinger inequalities (recall the proof of lemma 2.2) and that $g(0,\,0)=0$, we estimate

\begin{align*} 1& = \, \big\|g(u_{\infty}, \mathrm{D} u_{\infty}) \big\|_{L^{\infty}(\Omega)} \\ & \leq \, \mathrm{diam}(\Omega) \big\|\mathrm{D} (g(u_{\infty}, \mathrm{D} u_{\infty})) \big\|_{L^{\infty}(\Omega)} \\ & \leq \, \mathrm{diam}(\Omega)\Big( \big\|\partial_\eta g (u_{\infty}, \mathrm{D} u_{\infty})\mathrm{D} u_{\infty} \big\|_{L^{\infty}(\Omega)} + \big\| \partial_P g (u_{\infty}, \mathrm{D} u_{\infty})\mathrm{D}^2 u_{\infty} \big\|_{L^{\infty}(\Omega)} \Big) \\ & \leq \, \mathrm{diam}(\Omega)\Big( \big\|\partial_\eta g \big\|_{L^{\infty} ((u_{\infty}, \mathrm{D} u_{\infty})(\overline{\Omega}))}\|\mathrm{D} u_{\infty}\|_{L^{\infty}(\Omega)} \\ & \quad + \|\partial_P g \|_{L^{\infty} ((u_{\infty}, \mathrm{D} u_{\infty})(\overline{\Omega} ))}\|\mathrm{D}^2 u_{\infty}\|_{L^{\infty}(\Omega)}\Big) \\ & \leq \, \| \mathrm{D}^2 u_{\infty}\|_{L^{\infty}(\Omega)} \mathrm{diam}(\Omega) \Big( C(\infty,\Omega) \|\partial_\eta g \|_{L^{\infty}((u_{\infty}, \mathrm{D} u_{\infty})(\overline{\Omega} ))} \\ & \quad + \, \|\partial_P g \|_{L^{\infty}((u_{\infty}, \mathrm{D} u_{\infty})(\overline{\Omega})}\Big), \end{align*}

where $C(\infty,\,\Omega )>0$ is the maximum of the Poincaré and the Poincaré-Wirtinger inequality constants on $\Omega$ for $p=\infty$ (with the former being equal to $\mathrm {diam}(\Omega )$). As $g\geq 0$ and $\|g(u_{\infty },\, \mathrm {D} u_{\infty })\|_{L^{\infty }(\Omega )}=1$, we conclude that $0\leq g(u_{\infty },\, \mathrm {D} u_{\infty })\leq 1$ on $\overline {\Omega }$. Hence $(u_{\infty },\, \mathrm {D} u_{\infty })(\overline {\Omega }) \subseteq \{0\leq g \leq 1\}=\{g \leq 1\}$. Thus

\[ 1\leq \| \mathrm{D}^2 u_{\infty}\|_{L^{\infty}(\Omega)} \mathrm{diam}(\Omega) \Big( C(\infty,\Omega) \|\partial_\eta g \|_{L^{\infty}(\{ g \leq 1\})} + \|\partial_P g \|_{L^{\infty}(\{g \leq 1\})}\Big) \]

Rearranging assumption (1.3)(d), we may write $|X|\leq C_4^{-\frac {1}{\alpha }}(f(X)+C_3)^{\frac {1}{\alpha }}$, for any $X \in \mathbb {R}^{N \times n^2}_s$. Combining this inequality with the previous bound, we deduce

\begin{align*} C_4^{\frac{1}{\alpha}} & \leq \, \Big(\big\|f(\mathrm{D}^2 u_{\infty})\big\|_{L^{\infty}(\Omega)} +\, C_3\Big)^{\!\frac{1}{\alpha}} \mathrm{diam}(\Omega) \Big( C(\infty,\Omega) \|\partial_\eta g \|_{L^{\infty}(\{ g \leq 1\})} \\ & \quad + \, \|\partial_P g \|_{L^{\infty}(\{g \leq 1\})}\Big), \end{align*}

which leads directly to the claimed lower bound for the eigenvalue.

Now we establish the upper bound for $\Lambda _\infty$. Since $\Omega$ is by assumption a bounded domain with $C^2$ boundary, by standard results (see e.g. [Reference Gilbarg and Trudinger16, Sec. 14.6]), the distance function

\[ d_\Omega \equiv \mathrm{dist}({\cdot},\partial\Omega) : \mathbb{R}^n \longrightarrow \mathbb{R}, \]

which is in $W^{1,\infty }_{\mathrm {loc}}(\mathbb {R}^n)$, is also $C^2$ on an inner tubular neighbourhood of $\partial \Omega$, namely there exists $\varepsilon _0 \in (0,\,1)$ such that

\[ d_\Omega \in C^2(\overline{\Omega^{\varepsilon_0}}), \quad \Omega^\varepsilon := \{d_\Omega <\varepsilon\} \cap \Omega. \]

Let us also for convenience symbolize $\Omega _\varepsilon := \{d_\Omega >\varepsilon \} \cap \hspace {1pt} \Omega$. Let us also fix $k\in \{1,\,2\}$, a unit vector $e\in \mathbb {R}^N$ and $\zeta \in C^2(\mathbb {R}^n)$ with $\zeta \equiv 0$ on $\Omega _{\varepsilon _0}$. Then, for any $t_0>0$, the map $\xi _0 := t_0 (d_\Omega )^k\zeta e$ satisfies

\[ \xi_0 \, \in \, C^2(\overline{\Omega};\mathbb{R}^N). \]

Since $\mathrm {d}_\Omega =0$ on $\partial \Omega$ and also $\mathrm {D}(\mathrm {d}^2_\Omega )=0$ on $\partial \Omega$, it follows that $\xi _0 \in W^{2,\infty }_{\mathrm H}(\Omega ;\mathbb {R}^N)$ if $k=1$, whilst $\xi _0 \in W^{2,\infty }_{\mathrm C}(\Omega ;\mathbb {R}^N)$ if $k=2$. We will consider both cases simultaneously and declare this as

\[ \xi_0 \in W^{2,\infty}_{\mathrm B}(\Omega;\mathbb{R}^N). \]

By lemma 2.1, we can adjust the constant $t_0>0$ to arrange

\[ \big\| g(\xi_0,\mathrm{D} \xi_0)\big\|_{L^\infty(\Omega)} = 1. \]

Hence, $\xi _0$ is in the admissible class of the minimization problem (1.1). By minimality and assumption (1.3), we have the estimate

(2.3)\begin{equation} \Lambda_\infty \, \leq\, C_5 (t_0)^\alpha \Big(\big\| \mathrm{D}^2(d_\Omega^k \zeta)\big\|_{L^\infty(\Omega^{\varepsilon_0})}\Big)^\alpha +\, C_6. \end{equation}

By a direct computation, we have

(2.4)\begin{equation} \left\{ \begin{array}{@{}ll} \mathrm{D}^2(d_\Omega^k \zeta) \, & = \, k\big[ (k-1)\mathrm{D} d_\Omega \otimes \mathrm{D} d_\Omega + d_\Omega^{k-1}\mathrm{D}^2 d_\Omega \big] \zeta + d_\Omega^k \mathrm{D}^2 \zeta \\ & + \, k d_\Omega^{k-1} \big( \mathrm{D} d_\Omega\otimes \mathrm{D} \zeta + \mathrm{D} \zeta \otimes \mathrm{D} d_\Omega \big), \end{array} \right. \end{equation}

on $\overline {\Omega }$. For any $x\in \overline {\Omega ^{\varepsilon _0}}$, let us set $\mathrm {P}_{\Omega }(x) := \mathrm {Proj}_{\partial \Omega }(x)$. Then, by [Reference Gilbarg and Trudinger16, Sec. 14.6, L. 14.17], it follows that $|x- \mathrm {P}_{\Omega }(x)|=d_\Omega (x)$, and we also have the next estimates

(2.5)\begin{equation} \left\{ \begin{array}{@{}ll} \| d_\Omega \|_{L^\infty(\Omega^{\varepsilon_0})} & \leq \varepsilon_0, \phantom{\Big|} \\ \| \mathrm{D} d_\Omega \|_{L^\infty(\Omega^{\varepsilon_0})} & \leq 1, \\ \big\| \mathrm{D}^2 d_\Omega \big\|_{L^\infty(\Omega^{\varepsilon_0})} & \leq \sum_{i=1}^{n-1} \left\| \dfrac{ \kappa_i \circ \mathrm{P}_{\Omega} }{ 1 - ( \kappa_i \circ \mathrm{P}_{\Omega})d_\Omega } \right\|_{L^\infty(\Omega^{\varepsilon_0})}, \end{array} \right. \end{equation}

where $\{\kappa _1,\,...,\,\kappa _{n-1}\}$ are the principal curvatures of $\partial \Omega$. By (2.3)-(2.5) we have the estimate

(2.6)\begin{align} \Lambda_\infty & \leq\, C_5 (2t_0)^\alpha \Bigg(\| \mathrm{D}^2 \zeta \|_{L^\infty(\Omega)} + \| \mathrm{D} \zeta \|_{L^\infty(\Omega)} \nonumber\\ & \quad + \| \zeta \|_{L^\infty(\Omega)} \left( 1+ \sum_{i=1}^{n-1} \left\| \frac{ \kappa_i \circ \mathrm{P}_{\Omega} }{ 1 - ( \kappa_i \circ \mathrm{P}_{\Omega})d_\Omega } \right\|_{L^\infty(\Omega^{\varepsilon_0})} \right) \!\!\Bigg)^{\!\alpha} +\, C_6. \end{align}

It remains to select an appropriate function $\zeta$ in order to estimate its derivatives in terms of the geometry of $\Omega$, and to obtain an estimate for $t_0$. For the former, we argue as follows. Let $(\eta ^\delta )_{\delta >0}$ be the family of standard mollifying kernels, as e.g. in [Reference Katzourakis and Varvaruca27]. We select

\[ \zeta := \eta^{\varepsilon_0} * (\chi_{\mathbb{R}^n\setminus\Omega}), \]

which is the regularization of the characteristic of the complement of $\Omega$. It follows that this function satisfies the initial requirements, and additionally

\[ \left\{ \begin{array}{@{}ll} \mathrm{D} \zeta & = \eta^{\varepsilon_0} * (\mathrm{D} \chi_{\mathbb{R}^n\setminus\Omega}) = \eta^{\varepsilon_0} * \big(\mathcal{H}^{n-1}\text{$\llcorner$}_{\partial\Omega} \mathrm{D} \mathrm{d}_\Omega \big), \\ \mathrm{D}^2 \zeta & = \mathrm{D}\eta^{\varepsilon_0} * (\mathrm{D} \chi_{\mathbb{R}^n\setminus\Omega}) = \dfrac{1}{\varepsilon_0}(\mathrm{D}\eta)^{\varepsilon_0} * \big(\mathcal{H}^{n-1}\text{$\llcorner$}_{\partial\Omega} \mathrm{D} \mathrm{d}_\Omega \big), \end{array} \right. \]

by standard properties of convolutions and the differentiation of BV functions (see e.g. [Reference Folland15] and [Reference Evans and Gariepy14, Ch. 5, p. 198]). Then, by Young's inequality for convolutions, we have the estimates

(2.7)\begin{equation} \left\{ \begin{array}{@{}ll} \| \mathrm{D}^2 \zeta \|_{L^\infty(\mathbb{R}^n)} & \leq \dfrac{C}{\varepsilon_0^{n+1}}\mathcal{H}^{n-1}(\partial\Omega), \\ \| \mathrm{D} \zeta \|_{L^\infty(\mathbb{R}^n)} & \leq \mathcal{H}^{n-1}(\partial\Omega), \\ \| \zeta \|_{L^\infty(\mathbb{R}^n)} & \leq 1, \phantom{\Big|} \end{array} \right. \end{equation}

for some universal constant $C>0$. Now we work towards an estimate for $t_0$ appearing in (2.3). By assumption (1.4), we have that the sublevel sets $\{g\leq t\}$ are compact in $\mathbb {R}^N \times \mathbb {R}^{N\times n}$ for any $t\geq 0$. Let us define $R(t)$ as the smallest radius of the $N$-dimensional ball, for which $\{g\leq t\}$ is contained into the cylinder $\bar{\mathbb {B}}^N_{R(t)}(0) \times \mathbb {R}^{N\times n}$:

(2.8)\begin{equation} R(t) \,:=\, \inf \Big\{ R>0 \, : \, \{g\leq t\} \subseteq \mathbb{B}^N_R(0) \times \mathbb{R}^{N\times n}\Big\}. \end{equation}

Then, we define a strictly increasing function $\rho : [0,\,\infty ) \longrightarrow [0,\,\infty )$ by setting

(2.9)\begin{equation} \rho(t) \,:=\, t+\sup_{0\leq s\leq t}R(s). \end{equation}

Then, $\rho$ satisfies $\rho (0)=0$, and also that

\[ \{g\leq t\} \, \subseteq \, \bar{\mathbb{B}}^N_{\rho(t)}(0) \times \mathbb{R}^{N\times n}, \]

for any $t\geq 0$. Further, by construction,

\[ \Big\{(\eta, P) \in \mathbb{R}^N \times \mathbb{R}^{N\times n}\, :\ \rho^{{-}1}(|\eta|)\leq t \Big\} = \bar{\mathbb{B}}^N_{\rho(t)}(0) \times \mathbb{R}^{N\times n}. \]

The above imply

\[ \rho^{{-}1}(|\eta|) \,\leq\, g(\eta,P),\quad (\eta, P) \in \mathbb{R}^N \times \mathbb{R}^{N\times n}. \]

Next, since $d_\Omega ^k\zeta$ vanishes on $\partial \Omega \cup \overline {\Omega _{\varepsilon _0}}$ and $g(0,\,0)=0$, we have

\begin{align*} 1\, & =\, \|g(\xi_0,\mathrm{D}\xi_0)\|_{L^\infty(\Omega)} \\ & =\, \sup_{\Omega^{\varepsilon_0}}g(\xi_0,\mathrm{D}\xi_0) \\ & \geq\, \sup_{\Omega^{\varepsilon_0}}\rho^{{-}1}(|\xi_0|) \\ & \geq\, \sup_{\Omega^{\varepsilon_0}}\rho^{{-}1}\big(t_0 |d_\Omega^k \zeta|\big). \end{align*}

Since $d_\Omega \equiv \varepsilon _0/4$ on $\partial \Omega ^{\varepsilon _0/4}$, and $\rho ^{-1}$ is strictly increasing, the above implies

(2.10)\begin{align} 1\, & \geq\, \max_{\partial\Omega^{\varepsilon_0/4}}\rho^{{-}1}\big(t_0 |d_\Omega^k \zeta|\big) \nonumber\\ & =\, \max_{\partial\Omega^{\varepsilon_0/4}}\rho^{{-}1}\Big(t_0 \Big(\frac{\varepsilon_0}{4}\Big)^k \zeta\Big) \nonumber\\ & =\, \rho^{{-}1}\Big(t_0 \Big(\frac{\varepsilon_0}{4}\Big)^k \max_{\partial\Omega^{\varepsilon_0/4}}\zeta\Big). \end{align}

Now we estimate $\max _{\partial \Omega ^{\varepsilon _0/4}}\zeta$ from below. Fix $x\in \partial \Omega ^{\varepsilon _0/4}$. Then, since the standard mollifying kernel $\eta$ is a radial function (see e.g. [Reference Katzourakis and Varvaruca27]), there exists a universal $c>0$ such that $\eta \geq c$ on $\mathbb {B}_{1/2}(0)$. Therefore,

\begin{align*} \zeta(x) \, & =\, \frac{1}{\varepsilon_0^n}\int_{\mathbb{B}_{\varepsilon_0}(x)} \chi_{\mathbb{R}^n \setminus\Omega} \eta\Big( \frac{|y-x|}{\varepsilon_0}\Big) \mathrm d y \\ & \geq\, \frac{1}{\varepsilon_0^n}\int_{\mathbb{B}_{\varepsilon_0/2}(x) \setminus\Omega} \eta\Big( \frac{|y-x|}{\varepsilon_0}\Big) \mathrm d y \\ & \geq\, \frac{c}{\varepsilon_0^n}\mathcal{L}^n\big(\mathbb{B}_{\varepsilon_0/2}(x) \setminus\Omega \big), \end{align*}

for any $x\in \partial \Omega ^{\varepsilon _0/4}$. Finally, since $\partial \Omega$ satisfies the exterior sphere condition, the set $\mathbb {B}_{\varepsilon _0/2}(x) \setminus \Omega$ contains a ball $\mathbb {B}_r(\bar x)$ centred at some point $\bar x$, where the maximum possible radius $\bar r$ is given by

\[ \bar r = \min\bigg\{\frac{\varepsilon_0}{8} \,,\, \underset{i=1,...,n-1}{\min}\frac{1}{\|\kappa_i\|_{C^0(\partial\Omega)}} \bigg\} . \]

Therefore, if $\omega (n)$ is the volume of the unit ball in $\mathbb {R}^n$,

\begin{align*} \zeta(x) \, & \geq\, \frac{c}{\varepsilon_0^n}\mathcal{L}^n\big(\mathbb{B}_{\varepsilon_0/2}(x) \setminus\Omega \big) \\ & \geq\, \frac{c}{\varepsilon_0^n}\mathcal{L}^n(\mathbb{B}_{\bar r}(\bar x)) \\ & =\, \frac{c}{\varepsilon_0^n}\omega(n) \bar r^n \\ & =\, \frac{c \omega(n)}{\varepsilon_0^n} \min\bigg\{\Big(\frac{\varepsilon_0}{8}\Big)^n \,,\, \underset{i=1,...,n-1}{\min}\frac{1}{\big(\|\kappa_i\|_{C^0(\partial\Omega)}\big)^n} \bigg\} \\ & =\, c \omega(n) \min\bigg\{\frac{1}{2^{3n}} \,,\, \underset{i=1,...,n-1}{\min}\frac{1}{\big(\varepsilon_0\|\kappa_i\|_{C^0(\partial\Omega)}\big)^n} \bigg\}, \end{align*}

for any $x\in \partial \Omega ^{\varepsilon _0/4}$. Hence, we have established the lower bound

(2.11)\begin{equation} \max_{\partial\Omega^{\varepsilon_0/4}}\zeta \, \geq\, c \omega(n) \min\bigg\{\frac{1}{2^{3n}} \,,\, \underset{i=1,...,n-1}{\min}\frac{1}{\big(\varepsilon_0\|\kappa_i\|_{C^0(\partial\Omega)}\big)^n} \bigg\}. \end{equation}

By (2.10) and (2.11), we infer (since $\varepsilon _0<1$ and $k\in \{1,\,2\}$) that

(2.12)\begin{align} t_0 \, & \leq\, \frac{4^k \rho(1)\varepsilon_0^{n-k}}{c \omega(n)}\frac{1}{\min\bigg\{\dfrac{1}{2^{3n}}, \, \underset{i=1,...,n-1}{\min}\dfrac{1}{\big(\varepsilon_0\|\kappa_i\|_{C^0(\partial\Omega)}\big)^n} \bigg\}} \nonumber\\ & \leq\, \frac{32 \rho(1)}{c \omega(n)}\Big(2^{3n} + \underset{i=1,...,n-1}{\max}\big(\|\kappa_i\|_{C^0(\partial\Omega)}\big)^n \Big). \end{align}

By (2.6), (2.7), and (2.12), we conclude with the following upper bound for the eigenvalue:

(2.13)\begin{align} \Lambda_\infty \, & \leq\ C_6 + \, C_5 \left[ \frac{16\rho(1)}{c \omega(n)}\Big(2^{3n} + \, \underset{i=1,...,n-1}{\max}\big(\|\kappa_i\|_{C^0(\partial\Omega)}\big)^n \Big) \right]^\alpha \centerdot \nonumber\\ & \quad \centerdot \Bigg\{ 1 + \bigg(1+\frac{C}{\varepsilon_0^{n+1}} \bigg) \mathcal{H}^{n-1}(\partial\Omega) + \, \sum_{i=1}^{n-1} \left\| \frac{ \kappa_i \circ \mathrm{P}_{\Omega} }{ 1 - ( \kappa_i \circ \mathrm{P}_{\Omega})d_\Omega } \right\|_{L^\infty(\Omega^{\varepsilon_0})} \!\!\Bigg\}^{\!\alpha}. \end{align}

The claimed estimate (1.8) follows from (2.13) above, by recalling that in view of (2.8)–(2.9), we have

\[ \rho(1) = 1 + \sup_{0\leq t \leq 1}R(t), \]

and also that the last term of (2.13) is finite at least when

\[ \varepsilon_0 < \frac{1}{\underset{i=1,...,n-1}{\max} \|\kappa_i\|_{C^0(\partial\Omega)}}. \]

The result ensues.

Proof Proof of lemma 2.6

We begin by noting that since $g\,{\geq}\, 0$ and ${\|g(u_p,\, \mathrm {D} u_p)\|_{L^p(\Omega )}\,{=}\,1}$, in view of (1.12) we have the bound

\[ \|\nu_p\|(\overline{\Omega}) = \nu_p(\overline{\Omega}) = {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} g(u_p, \mathrm{D} u_p)^{p-1}\, \mathrm{d} \mathcal{L}^n\leq \bigg( \, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} g(u_p, \mathrm{D} u_p)^{p}\, \mathrm{d} \mathcal{L}^n \bigg)^{\frac{p-1}{p}}=1. \]

By the sequential weak$^*$ compactness of the space of Radon measures we can conclude that $\nu _{p} \overset{\, *_{\phantom{|}}}{{\smash{\, -\!\!\!\!-\!\!\!\!\rightharpoonup}}\, } \nu _\infty,\, \text {in } \mathcal {M}(\overline {\Omega })$ up to the passage to a further subsequence. Now we establish appropriate total variation bounds for the measure $\mathrm M_p$. Since $f\geq 0$, by the bounds of lemma 2.4 and assumption (1.3), we estimate (for sufficiently large $p$)

\begin{align*} \|{\mathrm M}_p\|(\overline{\Omega})\, & = \, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} \bigg( \frac{f(\mathrm{D}^2 u_p)}{\Lambda_p}\bigg)^{p-1}|\partial f(\mathrm{D}^2 u_p)| \, \mathrm{d} \mathcal{L}^n \\ & \leq \, \frac{1}{\Lambda^{p-1}_p} \, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}} f(\mathrm{D}^2 u_p)^{p-1} \Big( C_5 f(\mathrm{D}^2 u_p)^{\beta}+C_6\Big) \, \mathrm{d} \mathcal{L}^n \\ & = \, \frac{C_5}{\Lambda^{p-1}_p} \,{-\hspace{-10.5pt}\displaystyle\int_{\Omega}} f(\mathrm{D}^2 u_p)^{p-1+\beta} \, \mathrm{d} \mathcal{L}^n + \, \frac{C_6}{\Lambda^{p-1}_p}{-\hspace{-10.5pt}\displaystyle\int_{\Omega}}f(\mathrm{D}^2 u_p)^{p-1} \, \mathrm{d} \mathcal{L}^n. \end{align*}

Hence,

\begin{align*} \|{\mathrm M}_p\|(\overline{\Omega})\, & \leq \, \frac{C_5}{\Lambda^{p-1}_p}\bigg(\, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}}f(\mathrm{D}^2 u_p)^p \, \mathrm{d} \mathcal{L}^n \bigg)^{\frac{p-1+\beta}{p}} + \, \frac{C_6}{\Lambda^{p-1}_p}\bigg(\, {-\hspace{-10.5pt}\displaystyle\int_{\Omega}}f(\mathrm{D}^2 u_p)^{p} \, \mathrm{d} \mathcal{L}^n\bigg)^{\frac{p-1}{p}} \\ & = \, C_5\frac{(L_p)^{p-1+\beta}}{\Lambda_p^{p-1}} + \, C_6\frac{(L_p)^{p-1}}{\Lambda_p^{p-1}} \\ & = \, \bigg(\frac{L_p}{\Lambda_p}\bigg)^{p-1}\big(C_5L_p^{\beta}+C_6\big) \\ & \leq \, \bigg(\frac{C_8}{C_1}\bigg)^{1-\frac{1}{p}}\Big(C_5(\Lambda_{\infty}+1)^{\beta}+C_6 \Big). \end{align*}

The above bound allows to conclude that $\mathrm M_{p} \overset{\, *_{\phantom{|}}}{{\smash{\, -\!\!\!\!-\!\!\!\!\rightharpoonup}}\, } {\mathrm M}_\infty \text {in} \ \mathcal {M}(\overline {\Omega }; \mathbb {R}^{N \times n^2}_s)$, along perhaps a further subsequence of indices $(p_j)_1^\infty$ as $j\to \infty$.

Proof Proof of lemma 2.7

Fix a test function $\phi \in C^2_{\mathrm B}(\overline {\Omega };\mathbb {R}^N)$ and $p>n/\alpha +2$ by (1.12) we may rewrite the PDE system in (1.11) as follows

\[ \int_{\Omega} \mathrm{D}^2 \phi : \mathrm{d} \mathrm M_p = \ \Lambda_p \, \int_{\Omega} \Big( \partial_\eta g (u_p, \mathrm{D} u_p) \cdot \phi + \partial_P g (u_p, \mathrm{D} u_p) : \mathrm{D} \phi \Big)\, \mathrm{d} \nu_p, \]

Recall that, by proposition 2.5, we have $\Lambda _p\longrightarrow \Lambda _{\infty }$ and also $(u_p,\,\mathrm {D} u_p)\longrightarrow (u_{\infty },\,\mathrm {D} u_{\infty })$ uniformly on $\overline {\Omega }$, as $p_j\to \infty$. By assumption (1.4)(a), we have that $\partial _\eta g (u_p,\,\mathrm {D} u_p)\longrightarrow \partial _\eta g (u_{\infty },\, \mathrm {D} u_{\infty })$ and also $\partial _P g (u_p,\,\mathrm {D} u_p)\longrightarrow \partial _P g (u_{\infty },\, \mathrm {D} u_{\infty })$, both uniformly on $\overline {\Omega }$, as $p_j\to \infty$. The result ensues by invoking lemma 2.6, in conjunction with weak$^*$-strong continuity of the duality pairing $\mathcal {M}(\overline {\Omega })\times C(\overline {\Omega }) \longrightarrow \mathbb {R}$.