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Jack Thompson

Publications and source records attributed to Jack Thompson.

10 recordsLinked to original sources
arXiv2026

Volumetric density estimates for nonlocal minimal surfaces

In this article, we prove that viscosity subsolutions to nonlocal mean curvature-type equations satisfy universal volumetric estimates at all scales. Our results hold for general symmetric kernels that are comparable to the fractional Laplacian. Furthermore, we prove that subsolutions with low density (with respect to a universal constant) necessarily have `fat boundary', that is, have topological boundary with positive Lebesgue measure.

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arXiv2025

Stability results for nonlocal Serrin-type problems, antisymmetric Harnack inequalities, and geometric estimates

In this thesis, we explore several related topics broadly regarding the symmetry and geometric properties of nonlocal partial differential equations (PDE). 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 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. In the third part, we prove several geometric identities and inequalities involving the fractional mean curvature.

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arXiv2024

The nonlocal Harnack inequality for antisymmetric functions: an approach via Bochner's relation and harmonic analysis

We revisit a Harnack inequality for antisymmetric functions that has been recently established 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 paper 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.

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arXiv2024

Density estimates and the fractional Sobolev inequality for sets of zero $s$-mean curvature

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 $C$ 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.

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arXiv2023

Quantitative stability for the nonlocal overdetermined Serrin problem

We establish quantitative stability for the nonlocal Serrin overdetermined problem, via the method of the moving planes. Interestingly, our stability estimate is even better than those obtained so far in the classical setting (i.e., for the classical Laplacian) via the method of the 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 -- i.e., 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.

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arXiv2023

Quantitative stability for overdetermined nonlocal problems with parallel surfaces and investigation of the stability exponents

In this article, we analyze the stability of the parallel surface problem for semilinear equations driven by the fractional Laplacian. We prove a quantitative stability result that goes beyond that previously obtained in [Cir+23]. 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 [Cir+18] 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.

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arXiv2023

Some nonlocal formulas inspired by an identity of James Simon

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' work that does not heavily rely on differential geometry.

math.AP↗
arXiv2022

On the Harnack inequality for antisymmetric $s$-harmonic functions

We prove the Harnack inequality for antisymmetric $s$-harmonic functions, and more generally for solutions of fractional equations with zero-th 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 super-solutions and local boundedness for sub-solutions in both the interior and boundary case. En passant, we also obtain a new mean value formula for antisymmetric $s$-harmonic functions.

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arXiv2022

The role of antisymmetric functions in nonlocal equations

We prove a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms. As an application, we use such a Hopf-type lemma in combination with the method of moving planes to prove symmetry for the semilinear fractional parallel surface problem. That is, we prove that non-negative 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 counter-examples to these theorems when only `local' assumptions are imposed on the solutions.

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