In this thesis, we explore several related topics broadly regarding the symmetry and geometric properties of nonlocal partial differential equations (PDE). Nonlocal PDE, also known as integro-differential equations or fractional PDE, have become increasingly popular over the past few decades due to their ability to accurately model real-world phenomena. Nonlocal PDE arise naturally in the study of stochastic processes and can be viewed as generalisations of PDE which allow the possibility of long-range interactions or forces. Some examples where nonlocal PDE arise in the applied sciences are crystal dislocation, elasticity, ecology, mathematical finance, the surface quasigeostrophic equation in fluid mechanics, image processing, the physics of plasmas and flames, magnetised plasmas and quantum mechanics.

This thesis is split into three parts. In the first part, we study two overdetermined problems, namely Serrin’s problem and the parallel surface problem, driven by the fractional Laplacian.

In Chapter 2, we use a Hopf-type lemma for antisymmetric supersolutions to the Dirichlet problem for the fractional Laplacian with zeroth-order terms, in combination with the method of moving planes, to prove symmetry for the semilinear fractional parallel surface problem. That is, we prove that nonnegative solutions to semilinear Dirichlet problems for the fractional Laplacian in a bounded open set $\Omega \subset {\mathbb {R}}^n$ must be radially symmetric if one of their level surfaces is parallel to the boundary of $\Omega $ ; in turn, $\Omega $ must be a ball.

Furthermore, we discuss maximum principles and the Harnack inequality for antisymmetric functions in the fractional setting and provide counterexamples to these theorems when only ‘local’ assumptions are imposed on the solutions. The construction of these counterexamples relies on an approximation result that states that ‘all antisymmetric functions are locally antisymmetric and s-harmonic up to a small error’.

In Chapter 3, we analyse the stability of the parallel surface problem for semilinear equations driven by the fractional Laplacian. We prove a quantitative stability result that goes beyond the one previously obtained in [Reference Ciraolo, Dipierro, Poggesi, Pollastro and Valdinoci1].

Moreover, we discuss in detail several techniques and challenges in obtaining the optimal exponent in this stability result. In particular, this includes an upper bound on the exponent via an explicit computation involving a family of ellipsoids. We also sharply investigate a technique that was proposed in [Reference Ciraolo, Figalli, Maggi and Novaga2] to obtain the optimal stability exponent in the quantitative estimate for the nonlocal Alexandrov’s soap bubble theorem, obtaining accurate estimates to be compared with a new, explicit example.

In Chapter 4, we establish quantitative stability for the nonlocal Serrin overdetermined problem, via the method of moving planes. Interestingly, our stability estimate is even better than those obtained so far in the classical setting (that is, for the classical Laplacian) via the method of moving planes.

A crucial ingredient is the construction of a new antisymmetric barrier, which allows a unified treatment of the moving planes method. This strategy allows us to establish a new general quantitative nonlocal maximum principle for antisymmetric functions, leading to new quantitative nonlocal versions of both the Hopf lemma and the Serrin corner point lemma.

All these tools, that is, the new antisymmetric barrier, the general quantitative nonlocal maximum principle and the quantitative nonlocal versions of both the Hopf lemma and the Serrin corner point lemma, are of independent interest.

In the second part, we study the Harnack inequality for solutions to nonlocal PDE which are antisymmetric, that is, they have an odd symmetry with respect to reflections across some hyperplane. This topic has a strong motivation coming from proving quantitative stability estimates for nonlocal overdetermined problems which we explain in more detail in the introduction to this part.

In Chapter 6, we prove the Harnack inequality for antisymmetric s-harmonic functions, and more generally for solutions of fractional equations with zeroth-order terms, in a general domain. This may be used in conjunction with the method of moving planes to obtain quantitative stability results for symmetry and overdetermined problems for semilinear equations driven by the fractional Laplacian.

The proof is split into two parts: an interior Harnack inequality away from the plane of symmetry, and a boundary Harnack inequality close to the plane of symmetry. We prove these results by first establishing the weak Harnack inequality for supersolutions and local boundedness for subsolutions in both the interior and boundary case. En passant, we also obtain a new mean value formula for antisymmetric s-harmonic functions.

In Chapter 7, we revisit the Harnack inequality for antisymmetric functions that has been established in Chapter 6 for the fractional Laplacian and we extend it to more general nonlocal elliptic operators.

The new approach to deal with these problems that we propose in this chapter leverages Bochner’s relation, allowing one to relate a one-dimensional Fourier transform of an odd function with a three-dimensional Fourier transform of a radial function.

In this way, Harnack inequalities for odd functions, which are essentially Harnack inequalities of boundary type, are reduced to interior Harnack inequalities.

In the third part, we prove several geometric identities and inequalities involving the fractional mean curvature.

In Chapter 9, inspired by a classical identity proved by James Simons, we establish a new geometric formula in a nonlocal, possibly fractional, setting. Our formula also recovers the classical case in the limit, thus providing an approach to Simons’s work that does not rely heavily on differential geometry.

In Chapter 10, we prove that measurable sets $E\subset {\mathbb {R}}^n$ with locally finite perimeter and zero s-mean curvature satisfy the surface density estimates

$$ \begin{align*} \operatorname{Per} (E; B_R(x)) \geq CR^{n-1} \end{align*} $$

for all $R>0$ , $x\in \partial ^\ast E$ . The constant C depends only on n and s, and remains bounded as $s\to 1^-$ . As an application, we prove that the fractional Sobolev inequality holds on the boundary of sets with zero s-mean curvature.

Some of this research has been published in [Reference Dipierro, Poggesi, Thompson and Valdinoci3–Reference Dipierro, Thompson and Valdinoci5].