We consider the following problem

\begin{equation*}\varepsilon^{2 s} P_{\hat{h}}^s u+u=u^p, \quad u \gt 0 \quad \text {on } \,(M, \hat{h}),\end{equation*}

$(X, g^{+})$

$(M,[\hat{h}])$

$s\in (0,1)$

$P_{\hat{h}}^s$

$1 \lt p \lt \frac{n+2s}{n-2s}$

$\varepsilon$

$C^1$

$H$

$0 \lt s \lt {1}/{2}$

$C^1$

${1}/{2}\le s \lt 1$

We consider the existence of normalized solutions to non-linear Schrödinger equations on non-compact metric graphs in the L2-supercritical regime. For sufficiently small prescribed mass (L2 norm), we prove existence of positive solutions on two classes of graphs: periodic graphs and non-compact graphs with finitely many edges and suitable topological assumptions. Our approach is based on mountain pass techniques. A key point to overcome the serious lack of compactness is to show that all solutions with small mass have positive energy. To complement our analysis, we prove that this is no longer true, in general, for large masses. To the best of our knowledge, these are the first results with an L2-supercritical non-linearity extended on the whole graph and unravelling the role of topology in the existence of solutions.

In this paper, we are concerned with the following Dirichlet problems for nonlinear equations involving the fractional $p$-Laplacian:

\begin{equation*}\begin{cases}(-\Delta)_p^\alpha u=f(x,u,\nabla u),\ \ u \gt 0,\ \ \text{in}\ \ E,\\\ \ \ \ \ \ u\equiv0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{in}\ \ \mathbb{R}^{n}\setminus E,\end{cases}\end{equation*}

where $E \subseteq \mathbb{R}^{n}$ is a coercive epigraph, i.e., there exists a continuous function $\phi: \, \mathbb{R}^{n-1} \rightarrow \mathbb{R}$ satisfying

\begin{equation*}\lim_{|x'|\rightarrow+\infty}\phi(x')=+\infty,\end{equation*}

such that $E:=\{x=(x',x_{n}) \in \mathbb{R}^{n}|\,x_{n} \gt \phi(x')\}$, where $x':= (x_{1},...,x_{n-1}) \in \mathbb{R}^{n-1}$. Under some mild assumptions on the nonlinearity $f(x,u,\nabla u)$, we prove strict monotonicity of positive solutions to the above Dirichlet problems involving fractional $p$-Laplacian in coercive epigraph $E$.

Given $p\in[1,\infty)$ and a bounded open set $\Omega\subset\mathbb{R}^d$ with Lipschitz boundary, we study the $\Gamma$-convergence of the weighted fractional seminorm

\begin{equation*}[u]_{s,p,f}^p=\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{|\tilde u(x)- \tilde u(y)|^p}{\|x-y\|^{d+sp}}\,f(x)\,f(y)\,\mathrm{d} x\,\mathrm{d} y,\end{equation*}

as $s\to1^-$ for $u\in L^p(\Omega)$, where $\tilde u=u$ on $\Omega$ and $\tilde u=0$ on $\mathbb{R}^d\setminus\Omega$. Assuming that $(f_s)_{s\in(0,1)}\subset L^\infty(\mathbb{R}^d;[0,\infty))$ and $f\in\mathrm{Lip}_b(\mathbb{R}^d;(0,\infty))$ are such that $f_s\to f$ in $L^\infty(\mathbb{R}^d)$ as $s\to1^-$, we show that $(1-s)[u]_{s,p,f_s}^p$ $\Gamma$-converges to the Dirichlet $p$-energy weighted by $f^2$. In the case $p=2$, we also prove the convergence of the corresponding gradient flows.

This paper investigates sharp stability estimates for the fractional Hardy–Sobolev inequality:

\begin{align*}\ \ \ \ \ \ \ \ \ \mu_{s,t}\left(\mathbb{R}^N\right) \left(\int_{\mathbb{R}^N} \frac{|u|^{2^*_s(t)}}{|x|^t} \,\mathrm{d}x \right)^{\frac{2}{2^*_s(t)}} \leq \int_{\mathbb{R}^N} \left|(-\Delta)^{\frac{s}{2}} u \right|^2 \,\mathrm{d}x, \quad \text{for all } u \in \dot{H}^s\left(\mathbb{R}^N\right)\end{align*}

where $s \in (0,1)$, $0 \lt t \lt 2s$, $N \gt 2s$ is an integer, and $2^*_s(t) = \frac{2(N-t)}{N-2s}$. Here, $\mu_{s,t}\left(\mathbb{R}^N\right)$ represents the best constant in the inequality. The primary focus is on the quantitative stability results of the above inequality and the corresponding Euler–Lagrange equation near a positive ground-state solution. Additionally, a qualitative stability result is established for the Euler–Lagrange equation, offering a thorough characterization of the Palais–Smale sequences for the associated energy functional. These results generalize the sharp quantitative stability results for the classical Sobolev inequality in $\mathbb{R}^N$, originally obtained by Bianchi and Egnell [J. Funct. Anal. 1991] as well as the corresponding critical exponent problem in $\mathbb{R}^N$, explored by Ciraolo, Figalli, and Maggi [Int. Math. Res. Not. 2017] in the framework of fractional calculus.

We study the energy-critical half-wave maps equation:

$$\begin{align*}\partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u} \end{align*}$$

for $\mathbf {u} : [0, T) \times \mathbb {R} \to \mathbb {S}^2$. Our main result establishes the global existence and uniqueness of solutions for all rational initial data $\mathbf {u}_0 : \mathbb {R} \to \mathbb {S}^2$. This demonstrates global well-posedness for a dense subset within the scaling-critical energy space $ \dot {H}^{1/2}(\mathbb {R}; \mathbb {S}^2) $. Furthermore, we prove soliton resolution for a dense subset of initial data in the energy space with uniform bounds for all higher Sobolev norms $\dot {H}^s$ for $s> 0$.

Our analysis utilizes the Lax pair structure of the half-wave maps equation on Hardy spaces in combination with an explicit flow formula. Extending these results, we establish global well-posedness for rational initial data (along with a soliton resolution result) for a generalized class of matrix-valued half-wave maps equations with target spaces in the complex Grassmannians $ \mathsf {Gr}_k(\mathbb {C}^d) $. Notably, this includes the complex projective spaces $ \mathbb {CP}^{d-1} \cong \mathsf {Gr}_1(\mathbb {C}^d) $ thereby extending the classical case of the target $\mathbb {S}^2 \cong \mathbb {CP}^1 $.

The Cahn–Hilliard model with reaction terms can lead to situations in which no coarsening is taking place and, in contrast, growth and division of droplets occur which all do not grow larger than a certain size. This phenomenon has been suggested as a model for protocells, and a model based on the modified Cahn–Hilliard equation has been formulated. We introduce this equation and show the existence and uniqueness of solutions. Then, formally matched asymptotic expansions are used to identify a sharp interface limit using a scaling of the reaction term, which becomes singular when the interfacial thickness tends to zero. We compute planar solutions and study their stability under non-planar perturbations. Numerical computations for the suggested model are used to validate the sharp interface asymptotics. In addition, the numerical simulations show that the reaction terms lead to diverse phenomena such as growth and division of droplets in the obtained solutions, as well as the formation of shell-like structures.

In this paper we prove a general uniqueness result in the inverse boundary value problem for the weighted p-Laplace equation in the plane, with smooth weights. We also prove a uniqueness result in dimension 3 and higher, for real analytic weights that are subject to a smallness condition on one of their directional derivatives. Both results are obtained by linearizing the equation at a solution without critical points. This unknown solution is then recovered, together with the unknown weight.

In this paper, we consider blow-up of solutions to the Cauchy problem for the following fractional Nonlinear Schrödinger Equation (NLS),

\begin{equation*}\mathrm{i} \, \partial_t u=(-\Delta)^s u-|u|^{2 \sigma} u \quad \text{in} \,\, \mathbb{R} \times \mathbb{R}^N,\end{equation*}

where $N \geq 2$, $1/2 \lt s \lt 1$, and $0 \lt \sigma \lt 2s/(N-2s)$. In the mass critical and supercritical cases, we establish a criterion for blow-up of solutions to the problem for cylindrically symmetric data. The results extend the known ones with respect to blow-up of solutions to the problem for radially symmetric data.

Let $n\ge2$, $s\in(0,1)$, and $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. In this paper, we investigate the global (higher-order) Sobolev regularity of weak solutions to the fractional Dirichlet problem

\begin{equation*}\begin{cases}(-\Delta)^su=f \ \ & \text{in}\ \ \Omega,\\u=0 \ \ & \text{in}\ \ \mathbb{R}^n\setminus\Omega.\end{cases}\end{equation*}

Precisely, we prove that there exists a positive constant $\varepsilon\in(0,s]$ depending on n, s, and the Lipschitz constant of Ω such that, for any $t\in[\varepsilon,\min\{1+\varepsilon,2s\})$, when $f\in L^q(\Omega)$ with some $q\in(\frac{n}{2s-t},\infty]$, the weak solution u satisfies

\begin{equation*}\|u\|_{W^{t,p}(\mathbb{R}^n)}\le C\|f\|_{L^q(\Omega)}\end{equation*}

for all $p\in[1,\frac{1}{t-\varepsilon})$. In particular, when Ω is a bounded C1 domain or a bounded Lipschitz domain satisfying the uniform exterior ball condition, the aforementioned global regularity estimates hold with $\varepsilon=s$ and they are sharp in this case. Moreover, if Ω is a bounded $C^{1,\kappa}$ domain with $\kappa\in(0,s)$ or a bounded Lipschitz domain satisfying the uniform exterior ball condition, we further show the global BMO-Sobolev regularity estimate

\begin{equation*}\left\|(-\Delta)^{\frac{s}{2}}u\right\|_{\mathrm{BMO}(\mathbb{R}^n)}+\left\|\nabla^{s}u\right\|_{\mathrm{BMO}(\mathbb{R}^n)}\le C\|f\|_{L^q(\Omega)}\end{equation*}

for some $q\in(\frac{n}{s},\infty]$, which is sharp in the sense that the BMO norm can not be improved to the $L^\infty$ norm.

In the second part of this series of papers, we address the same evolution problem that was considered in part 1 (see [16]), namely the nonlocal Fisher-KPP equation in one spatial dimension,

\begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*}

$\phi *u$

$\phi (y) \equiv H\left (\frac {1}{4}-y^2\right )$

$[0,a]$

The aim of this paper is to establish the correspondence between the twisted localised Pestov identity on the unit tangent bundle of a Riemannian manifold and the Weitzenböck identity for twisted symmetric tensors on the manifold.

Optimal transport tasks naturally arise in gas networks, which include a variety of constraints such as physical plausibility of the transport and the avoidance of extreme pressure fluctuations. To define feasible optimal transport plans, we utilize a $p$-Wasserstein metric and similar dynamic formulations minimizing the kinetic energy necessary for moving gas through the network, which we combine with suitable versions of Kirchhoff’s law as the coupling condition at nodes. In contrast to existing literature, we especially focus on the non-standard case $p \neq 2$ to derive an overdamped isothermal model for gases through $p$-Wasserstein gradient flows in order to uncover and analyze underlying dynamics. We introduce different options for modelling the gas network as an oriented graph including the possibility to store gas at interior vertices and to put in or take out gas at boundary vertices.

As is well-known, the conformal class of a surface M with boundary is determined by its Diriclet-to-Neumann (DN) map $\Lambda $. We propose an algorithm for determination of the b-period matrix $\mathbb {B}$ of the (Schottky) double of M via $\Lambda $.

In this paper, we establish suitable characterisations for a pair of functions $(W(x),H(x))$ on a bounded, connected domain $\Omega \subset \mathbb{R}^n$ in order to have the following Hardy inequality:

\begin{equation*}\int_{\Omega} W(x) |\nabla u|_A^2 dx \geq \int_{\Omega} |\nabla d|^2_AH(x)|u|^2 dx, \,\,\, u \in C^{1}_0(\Omega),\end{equation*}

where d(x) is a suitable quasi-norm (gauge), $|\xi|^2_A = \langle A(x)\xi, \xi \rangle$ for $\xi \in \mathbb{R}^n$ and A(x) is an n × n symmetric, uniformly positive definite matrix defined on a bounded domain $\Omega \subset \mathbb{R}^n$. We also give its Lp analogue. As a consequence, we present examples for a standard Laplacian on $\mathbb{R}^n$, Baouendi–Grushin operator, and sub-Laplacians on the Heisenberg group, the Engel group and the Cartan group. Those kind of characterisations for a pair of functions $(W(x),H(x))$ are obtained also for the Rellich inequality. These results answer the open problems of Ghoussoub-Moradifam [16].

We solve a problem posed by Calabi more than 60 years ago, known as the Saint-Venant compatibility problem: Given a compact Riemannian manifold, generally with boundary, find a compatibility operator for Lie derivatives of the metric tensor. This problem is related to other compatibility problems in mathematical physics, and to their inherent gauge freedom. To this end, we develop a framework generalizing the theory of elliptic complexes for sequences of linear differential operators $(A_{\bullet })$ between sections of vector bundles. We call such a sequence an elliptic pre-complex if the operators satisfy overdetermined ellipticity conditions and the order of $A_{k+1}A_k$ does not exceed the order of $A_k$. We show that every elliptic pre-complex $(A_{\bullet })$ can be ‘corrected’ into a complex $({\mathcal {A}}_{\bullet })$ of pseudodifferential operators, where ${\mathcal {A}}_k - A_k$ is a zero-order correction within this class. The induced complex $({\mathcal {A}}_{\bullet })$ yields Hodge-like decompositions, which in turn lead to explicit integrability conditions for overdetermined boundary-value problems, with uniqueness and gauge freedom clauses. We apply the theory on elliptic pre-complexes of exterior covariant derivatives of vector-valued forms and double forms satisfying generalized algebraic Bianchi identities, thus resolving a set of compatibility and gauge problems, among which one is the Saint-Venant problem.

We consider stochastic differential equations (SDEs) driven by a fractional Brownian motion with a drift coefficient that is allowed to be arbitrarily close to criticality in a scaling sense. We develop a comprehensive solution theory that includes strong existence, path-by-path uniqueness, existence of a solution flow of diffeomorphisms, Malliavin differentiability and $\rho $-irregularity. As a consequence, we can also treat McKean-Vlasov, transport and continuity equations.

Let $\Omega \subset \mathbb{R}^d$ with $d\geq 2$ be a bounded domain of class ${\mathcal C}^{1,\beta }$ for some $\beta \in (0,1)$. For $p\in (1, \infty )$ and $s\in (0,1)$, let $\Lambda ^s_{p}(\Omega )$ be the first eigenvalue of the mixed local–nonlocal operator $-\Delta _p+(-\Delta _p)^s$ in Ω with the homogeneous nonlocal Dirichlet boundary condition. We establish a strict Faber–Krahn-type inequality for $\Lambda _{p}^s(\cdot )$ under polarization. As an application of this strict inequality, we obtain the strict monotonicity of $\Lambda _{p}^s(\cdot )$ over the annular domains and characterize the rigidity property of the balls in the classical Faber–Krahn inequality for $-\Delta _p+(-\Delta _p)^s$.

Let X be a complex Banach space and B be a closed linear operator with domain $\mathcal{D}(B) \subset X,\,\, a,b,c,d\in\mathbb{R},$ and $0 \lt \beta \lt \alpha.$ We prove that the problem

\begin{equation*}u(t) -(aB+bI)(g_{\alpha-\beta}\ast u)(t) -(cB+dI)(g_{\alpha}\ast u)(t) = h(t), \quad t\geq 0,\end{equation*}

where $g_{\alpha}(t)=t^{\alpha-1}/\Gamma(\alpha)$ and $h:\mathbb{R}_+\to X$ is given, has a unique solution for any initial condition on $\mathcal{D}(B)\times X$ as long as the operator B generates an ad-hoc Laplace transformable and strongly continuous solution family $\{R_{\alpha,\beta}(t)\}_{t\geq 0} \subset \mathcal{L}(X).$ It is shown that such a solution family exists whenever the pair $(\alpha,\beta)$ belongs to a subset of the set $(1,2]\times(0,1]$ and B is the generator of a cosine family or a C0-semigroup in $X.$ In any case, it also depends on certain compatibility conditions on the real parameters $a,b,c,d$ that must be satisfied.