arXiv subjects

Marvin Weidner

Publications and source records attributed to Marvin Weidner.

At least 19 recordsLinked to original sources

arXiv2026

Capacitary estimates for solutions to nonlocal Dirichlet problems

We study the boundary regularity of weak solutions to nonlocal nonlinear elliptic equations with bounded measurable coefficients. Our main result establishes that a capacity density condition is equivalent to the validity of a uniform boundary H\"older estimate for solutions with H\"older continuous exterior data. More generally, we derive a fine capacitary estimate on the modulus of continuity that captures how regularity is inherited from the exterior datum to the solution.

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arXiv2026

Kinetic Fokker-Planck equations with Maxwell boundary conditions

We develop the boundary regularity theory for solutions to linear kinetic Fokker-Planck equations with Maxwell boundary conditions. These conditions interpolate between diffuse and specular reflection via an accommodation coefficient $α\in [0,1]$. While existing literature is restricted to the extreme cases $α= 0$ and $α= 1$, we resolve the entire intermediate regime $α\in (0,1)$. Specifically, we show that solutions are Hölder continuous if the coefficients are merely uniformly elliptic. Furthermore, for sufficiently smooth coefficients, we establish boundary regularity of order $\frac{3}π \arccos(\fracα{2}) - 1$ up to the grazing set. This exponent is optimal. Beyond Maxwell conditions, we develop a unified approach that extends to a broad class of reflection boundary conditions, including super-elastic collisions.

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arXiv2026

Boundary Harnack estimates of optimal order for kinetic Fokker-Planck equations

We establish higher order boundary Harnack estimates for solutions to kinetic Fokker-Planck equations with absorbing incoming boundaries. Unlike classical elliptic and parabolic equations with Dirichlet data, we show that the quotient of two solutions for kinetic equations is not $C^{\infty}$ up to the boundary. Instead, we develop a general theory showing that, near the grazing set, the quotient of two solutions is $C^{3/2}$ if the domain and data are sufficiently smooth, and $C^{1,1}$ in the absence of source terms. These exponents are optimal.

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arXiv2026

Sharp regularity near the grazing set for kinetic Fokker-Planck equations

We prove optimal regularity results for solutions to linear kinetic Fokker-Planck equations in bounded domains. Our contributions are two-fold. First, we establish the sharp $C^{1/2}$ regularity for either diffuse reflection or prescribed in-flow boundary conditions. Previously, in this setting, it was only known that solutions are $C^α$ for some small $α> 0$. Second, we provide a complete characterization of the solution behavior near the grazing set by proving higher order expansions beyond the critical regularity threshold of $\frac{1}{2}$. These results demonstrate for the first time that solutions maintain higher smoothness up to the grazing set near the incoming boundary.

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arXiv2025

Dirichlet heat kernel estimates for parabolic nonlocal equations

In this article we establish the optimal $C^s$ boundary regularity for solutions to nonlocal parabolic equations in divergence form in $C^{1,α}$ domains and prove a higher order boundary Harnack principle in this setting. Our approach applies to a broad class of nonlocal operators with merely Hölder continuous coefficients, but our results are new even in the translation invariant case. As an application, we obtain sharp two-sided estimates for the associated Dirichlet heat kernel. Notably, our estimates cover nonlocal operators with time-dependent coefficients, which had remained open in the literature.

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arXiv2025

The Neumann problem for the fractional Laplacian: optimal regularity via the Mellin transform

We establish the optimal regularity of solutions to the Neumann problem for the fractional Laplacian, $(-Δ)^s u=h$ in $Ω$, with the external condition $\mathcal N^s u=0$ in $Ω^c$. For this, a key point is to establish a 1D Liouville theorem for functions with growth, which we prove by using complex analysis and the Mellin transform. More precisely, we prove a ``meta-theorem'' relating the classification of 1D solutions to general linear homogeneous equations of the type $Lu=0$ in $(0,\infty)$ to the (complex) roots of an explicit meromorphic function $f(z)$ that depends on $L$. In case of the fractional Laplacian with Neumann conditions, we show that all solutions are $C^{2s+α}$ when $s\leq 1/2$, and $C^{s+\frac12+α}$ when $s\geq1/2$. Moreover, quite surprisingly, we prove that even in 1D there exist highly oscillating solutions of the type $u(x)=x^{a} \cos(b \log x)$ for $x>0$, with $a>0$ and $b>0$ that depend on $s$, and $a<2s$ for $s\sim1$.

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arXiv2025

Optimal boundary regularity and Green function estimates for nonlocal equations in divergence form

In this article we prove for the first time the $C^s$ boundary regularity for solutions to nonlocal elliptic equations with Hölder continuous coefficients in divergence form in $C^{1,α}$ domains. So far, it was only known that solutions are Hölder continuous up to the boundary, and establishing their optimal regularity has remained an open problem in the field. Our proof is based on a delicate higher order Campanato-type iteration at the boundary, which we develop in the context of nonlocal equations and which is quite different from the local theory. As an application of our results, we establish sharp two-sided Green function estimates in $C^{1,α}$ domains for the same class of operators. Previously, this was only known under additional structural assumptions on the coefficients and in more regular domains.

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arXiv2025

Higher regularity in nonlocal free boundary problems

We study the higher regularity in nonlocal free boundary problems posed for general integro-differential operators of order $2s$. Our main result is for the nonlocal one-phase (Bernoulli) problem, for which we establish that $C^{2,α}$ free boundaries are $C^\infty$. This is new even for the fractional Laplacian, as it was only known in case $s=\frac12$. We also establish a general result for overdetermined problems, showing that if the boundary condition is smooth, then so is $\partialΩ$. Our approach is very robust and works as well for the nonlocal obstacle problem, where it yields a new proof of the higher regularity of free boundaries, completely different from the one in [AbRo20]. In order to prove our results, we need to develop, among other tools, new integration by parts formulas and delicate boundary Hölder estimates for nonlocal equations with (local) Neumann boundary conditions that had not been studied before and are of independent interest.

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arXiv2025

Regularity for the Boltzmann equation conditional to pressure and moment bounds

We prove that solutions to the Boltzmann equation without cut-off satisfying pointwise bounds on some observables (mass, pressure, and suitable moments) enjoy a uniform bound in $L^\infty$ in the case of hard potentials. As a consequence, we derive $C^{\infty}$ estimates and decay estimates for all derivatives, conditional to these macroscopic bounds. Our $L^\infty$ estimates are uniform in the limit $s \nearrow 1$ and hence we recover the same results also for the Landau equation.

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arXiv2025

Optimal regularity for kinetic Fokker-Planck equations in domains

We study the smoothness of solutions to linear kinetic Fokker-Planck equations in domains $Ω\subset \mathbb{R}^n$ with specular reflection condition, including Kolmogorov's equation $\partial_t f +v\cdot\nabla_x f-Δ_v f=h$. Our main results establish the following: - Solutions are always $C^\infty$ in $t,v,x$ away from the grazing set $\{x\in\partialΩ,\ v\cdot n_x=0\}$. - They are $C^{4,1}_{\text{kin}}$ up to the grazing set. - This regularity is optimal, i.e. we show that that they are in general not $C^5_{\text{kin}}$. These results show for the first time that solutions are classical up to boundary, i.e. $C^1_{t,x}$ and $C^2_v$.

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arXiv2025

Time-insensitive nonlocal parabolic Harnack estimates

We establish new Harnack estimates that defy the waiting-time phenomenon for global solutions to nonlocal parabolic equations. Our technique allows us to consider general nonlocal operators with bounded measurable coefficients. Moreover, we show that a waiting-time is required for the nonlocal parabolic Harnack inequality when local solutions are considered.

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arXiv2025

Nonnegative solutions to nonlocal parabolic equations

We aim to study nonnegative, global solutions to a general class of nonlocal parabolic equations with bounded measurable coefficients. First, we prove a Widder-type theorem. Such a result has previously been studied only for certain translation invariant operators, and new ideas are needed in our general setting. Second, we establish sharp two-sided bounds for the fundamental solution via purely variational techniques, entirely bypassing tools from semigroup theory, Dirichlet forms, and stochastic analysis. Third, we derive sharp Harnack-type estimates that are novel even for the fractional heat equation.

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arXiv2025

Regularity for nonlocal equations with local Neumann boundary conditions

In this article we establish fine results on the boundary behavior of solutions to nonlocal equations in $C^{k,γ}$ domains which satisfy local Neumann conditions on the boundary. Such solutions typically blow up at the boundary like $v \asymp d^{s-1}$ and are sometimes called large solutions. In this setup we prove optimal regularity results for the quotients $v/d^{s-1}$, depending on the regularity of the domain and on the data of the problem. The results of this article will be important in a forthcoming work on nonlocal free boundary problems.

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arXiv2025

Improvement of flatness for nonlocal free boundary problems

In this article we study for the first time the regularity of the free boundary in the one-phase free boundary problem driven by a general nonlocal operator. Our main results establish that the free boundary is $C^{1,α}$ near regular points, and that the set of regular free boundary points is open and dense. Moreover, in 2D we classify all blow-up limits and prove that the free boundary is $C^{1,α}$ everywhere. The main technical tool of our proof is an improvement of flatness scheme, which we establish in the general framework of viscosity solutions, and which is of independent interest. All of these results were only known for the fractional Laplacian, and are completely new for general nonlocal operators. In contrast to previous works on the fractional Laplacian, our method of proof is purely nonlocal in nature.

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arXiv2024

The Harnack inequality fails for nonlocal kinetic equations

We prove that the Harnack inequality fails for nonlocal kinetic equations. Such equations arise as linearized models for the Boltzmann equation without cutoff and are of hypoelliptic type. We provide a counterexample for the simplest equation in this theory, the fractional Kolmogorov equation. Our result reflects a purely nonlocal phenomenon since the Harnack inequality holds true for local kinetic equations like the Kolmogorov equation.

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arXiv2024

Semiconvexity estimates for nonlinear integro-differential equations

In this paper we establish for the first time local semiconvexity estimates for fully nonlinear equations and for obstacle problems driven by integro-differential operators with general kernels. Our proof is based on the Bernstein technique, which we develop for a natural class of nonlocal operators and consider to be of independent interest. In particular, we solve an open problem from Cabré-Dipierro-Valdinoci [CDV22]. As an application of our result, we establish optimal regularity estimates and smoothness of the free boundary near regular points for the nonlocal obstacle problem on domains. Finally, we also extend the Bernstein technique to parabolic equations and nonsymmetric operators.

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arXiv2024

Optimal regularity for nonlocal elliptic equations and free boundary problems

In this article we establish for the first time the $C^s$ boundary regularity of solutions to nonlocal elliptic equations with kernels $K(y)\asymp |y|^{-n-2s}$. This was known to hold only when $K$ is homogeneous, and it is quite surprising that it holds for general inhomogeneous kernels, too. As an application of our results, we also establish the optimal $C^{1+s}$ regularity of solutions to obstacle problems for general nonlocal operators with kernels $K(y)\asymp |y|^{-n-2s}$. Again, this was only known when $K$ is homogeneous, and it solves a long-standing open question in the field. A new key idea is to construct a 1D solution as a minimizer of an appropriate nonlocal one-phase free boundary problem, for which we establish optimal $C^s$ regularity and non-degeneracy estimates.

math.AP↗

arXiv2024

The parabolic Harnack inequality for nonlocal equations

We complete the local regularity program for weak solutions to linear parabolic nonlocal equations with bounded measurable coefficients. Within the variational framework we prove the parabolic Harnack inequality and Hölder regularity estimates. We discuss in detail the shortcomings of previous results in this direction. The key element of our approach is a fine study of the nonlocal tail term.

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