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Tobias Weth

Publications and source records attributed to Tobias Weth.

At least 37 records · Page 2Linked to original sources
arXiv2022

On an electrostatic problem and a new class of exceptional subdomains of $\mathbb{R}^3$

We study the existence of nontrivial unbounded surfaces $S\subset \mathbb{R}^3$ with the property that the constant charge distribution on $S$ is an electrostatic equilibrium, i.e. the resulting electrostatic force is normal to the surface at each point on $S$. Among bounded regular surfaces $S$, only the round sphere has this property by a result of Reichel $[23]$ (see also Mendez and Reichel $[16]$) confirming a conjecture of P. Gruber. In the present paper, we show the existence of nontrivial exceptional domains $Ω\subset \mathbb{R}^3$ whose boundaries $S=\partial Ω$ enjoy the above property.

math.AP↗
arXiv2022

Short time existence and smoothness of the nonlocal mean curvature flow of graphs

We consider the geometric evolution problem of entire graphs moving by fractional mean curvature. For this, we study the associated nonlocal quasilinear evolution equation satisfied by the family of graph functions. We establish, using an analytic semigroup approach, short time existence, uniqueness and optimal Hölder regularity in time and space of classical solutions of the nonlocal equation, depending on the regularity of the initial graph. The method also yields $C^\infty-$smoothness estimates of the evolving graphs for positive times.

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arXiv2022

Rotating waves in nonlinear media and critical degenerate Sobolev inequalities

We investigate the presence of rotating wave solutions of the nonlinear wave equation $\partial_t^2 v - Δv +m v = |v|^{p-2} v$ in $\mathbb{R} \times \mathbf{B}$, where $\mathbf{B} \subset \mathbb{R}^N$ is the unit ball, complemented with Dirichlet boundary conditions on $\mathbb{R} \times \partial\mathbf{B}$. Depending on the prescribed angular velocity $α$ of the rotation, this leads to a Dirichlet problem for a semilinear elliptic or degenerate elliptic equation. We show that this problem is governed by an associated critical degenerate Sobolev inequality in the half space. After proving this inequality and the existence of associated extremal functions, we then deduce necessary and sufficient conditions for the existence of ground state solutions. Moreover, we analyze under which conditions on $α$, $m$ and $p$ these ground states are nonradial and therefore give rise to truly rotating waves. Our approach carries over to the corresponding Dirichlet problems in an annulus and in more general Riemannian models with boundary, including the hemisphere. We briefly discuss these problems and show that they are related to a larger family of associated critical degenerate Sobolev inequalities.

math.AP↗
arXiv2022

A fractional Hadamard formula and applications

We consider the domain dependence of the best constant in the subcritical fractional Sobolev constant, $$ λ_{s,p}(Ω):=\inf \left\{ [u]_{H^s(\mathbb{R}^N)}^2,\,\, u\in C^\infty_c(Ω),\,\, \|u\|_{L^p(Ω)}=1 \right\}, $$ where $s\in (0,1)$, $Ω$ is bounded of class $C^{1,1}$ and $p\in [1, \frac{2N}{N-2s})$ if $2s<N$, $p\in [1, \infty)$ if $2s\geq N=1$. Explicitly, we derive formula for the one-sided shape derivative of the mapping $Ω\mapsto λ_{s,p}(Ω)$ under domain perturbations. In the case where $ λ_{s,p}(Ω)$ admits a unique positive minimizer (e.g. $p=1$ or $p=2$), our result implies a nonlocal version of the classical variational Hadamard formula for the first eigenvalue of the Dirichlet Laplacian on $Ω$. Thanks to the formula for our one-sided shape derivative, we characterize smooth local minimizers of $λ_{s,p}(Ω)$ under volume-preserving deformations, and we find that they are balls if $p\in \{1\}\cup [2,\infty)$. Finally, we consider the maximization problem for $λ_{s,p}(Ω)$ among annular-shaped domains of fixed volume of the type $B\setminus \overline B'$, where $B$ is a fixed ball and $B'$ is ball whose position is varied within $B$. We prove that, for $p\in \{1,2\}$, the value $λ_{s,p}(B\setminus \overline B')$ is maximal when the two balls are concentric.

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arXiv2021

A connection between symmetry breaking for Sobolev minimizers and stationary Navier-Stokes flows past a circular obstacle

Fluid flows around a symmetric obstacle generate vortices which may lead to symmetry breaking of the streamlines. We study this phenomenon for planar viscous flows governed by the stationary Navier-Stokes equations with constant inhomogeneous Dirichlet boundary data in a rectangular channel containing a circular obstacle. In such (symmetric) framework, symmetry breaking is strictly related to the appearance of multiple solutions. Symmetry breaking properties of some Sobolev minimizers are studied and explicit bounds on the boundary velocity (in terms of the length and height of the channel) ensuring uniqueness are obtained after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the boundary data. We show that, regardless of the solenoidal extension employed, such bounds converge to zero at an optimal rate as the length of the channel tends to infinity.

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arXiv2021

A generalized fractional Pohozaev identity and applications

We prove a fractional Pohozaev type identity in a generalized framework and discuss its applications. Specifically, we shall consider applications to nonexistence of solutions in the case of supercritical semilinear Dirichlet problems and regarding a Hadamard formula for the derivative of Dirichlet eigenvalues of the fractional Laplacian with respect to domain deformations. We also derive the simplicity of radial eigenvalues in the case of radial bounded domains and apply the Hadamard formula to this case.

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arXiv2021

Existence of nonradial domains for overdetermined and isoperimetric problems in nonconvex cones

In this work we address the question of the existence of nonradial domains inside a nonconvex cone for which a mixed boundary overdetermined problem admits a solution. Our approach is variational, and consists in proving the existence of nonradial minimizers, under a volume constraint, of the associated torsional energy functional. In particular we give a condition on the domain $D$ on the sphere spanning the cone which ensures that the spherical sector is not a minimizer. Similar results are obtained for the relative isoperimetric problem in nonconvex cones.

math.AP↗
arXiv2021

Symmetry breaking for ground states of biharmonic NLS via Fourier extension estimates

We consider ground states solutions $u \in H^2(\mathbb{R}^N)$ of biharmonic (fourth-order) nonlinear Schrödinger equations of the form $$ Δ^2 u + 2a Δu + b u - |u|^{p-2} u = 0 \quad \mbox{in $\mathbb{R}^N$} $$ with positive constants $a, b > 0$ and exponents $2 < p < 2^*$, where $2^* = \frac{2N}{N-4}$ if $N > 4$ and $2^* = \infty$ if $N \leq 4$. By exploiting a connection to the adjoint Stein--Tomas inequality on the unit sphere and by using trial functions due to Knapp, we prove a general symmetry breaking result by showing that all ground states $u\in H^2(\mathbb{R}^N)$ in dimension $N \geq 2$ fail to be radially symmetric for all exponents $2 < p < \frac{2N+2}{N-1}$ in a suitable regime of $a,b>0$. As applications of our main result, we also prove symmetry breaking for a minimization problem with constrained $L^2$-mass and for a related problem on the unit ball in $\mathbb{R}^N$ subject to Dirichlet boundary conditions.

math.AP↗
arXiv2021

Symmetry properties of sign-changing solutions to nonlinear parabolic equations in unbounded domains

We study the asymptotic (in time) behavior of positive and sign-changing solutions to nonlinear parabolic problems in the whole space or in the exterior of a ball with Dirichlet boundary conditions. We show that, under suitable regularity and stability assumptions, solutions are asymptotically (in time) foliated Schwarz symmetric, i.e., all elements in the associated omega-limit set are axially symmetric with respect to a common axis passing through the origin and are nonincreasing in the polar angle. We also obtain symmetry results for solutions of Hénon-type problems, for equilibria (i.e. for solutions of the corresponding elliptic problem), and for time periodic solutions.

math.AP↗
arXiv2021

Morse index versus radial symmetry for fractional Dirichlet problems

In this work, we provide an estimate of the Morse index of radially symmetric sign changing bounded weak solutions $u$ to the semilinear fractional Dirichlet problem $$ (-Δ)^su = f(u)\qquad \text{ in $\mathcal{B}$},\qquad \qquad u = 0\qquad \text{in $\quad\mathbb{R}^{N}\setminus \mathcal{B}$,} $$ where $s\in(0,1)$, $\mathcal{B}\subset \mathbb{R}^N$ is the unit ball centred at zero and the nonlinearity $f$ is of class $C^1$. We prove that for $s\in(1/2,1)$ any radially symmetric sign changing solution of the above problem has a Morse index greater than or equal to $N+1$. If $s\in (0,1/2],$ the same conclusion holds under additional assumption on $f$. In particular, our results apply to the Dirichlet eigenvalue problem for the operator $(-Δ)^s$ in $\mathcal{B}$ for all $s\in (0,1)$, and it implies that eigenfunctions corresponding to the second Dirichlet eigenvalue in $\mathcal{B}$ are antisymmetric. This resolves a conjecture of Bañuelos and Kulczycki.

math.AP↗
arXiv2021

Complex solutions and stationary scattering for the nonlinear Helmholtz equation

We study a stationary scattering problem related to the nonlinear Helmholtz equation $-Δu - k^2 u = f(x,u) \ \ \text{in $\mathbb{R}^N$,}$ where $N \ge 3$ and $k>0$. For a given incident free wave $φ\in L^\infty(\mathbb{R}^N)$, we prove the existence of complex-valued solutions of the form $u=φ+u_{\text{sc}}$, where $u_{\text{sc}}$ satisfies the Sommerfeld outgoing radiation condition. Since neither a variational framework nor maximum principles are available for this problem, we use topological fixed point theory and global bifurcation theory to solve an associated integral equation involving the Helmholtz resolvent operator. The key step of this approach is the proof of suitable a priori bounds.

math.AP↗
arXiv2021

Small order asymptotics of the Dirichlet eigenvalue problem for the fractional Laplacian

In this article, we study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian $(-Δ)^s$ in bounded open Lipschitz sets in the small order limit $s \to 0^+$. While it is easy to see that all eigenvalues converge to $1$ as $s \to 0^+$, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with symbol $2\log|ξ|$. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that $L^2$-normalized Dirichlet eigenfunctions of $(-Δ)^s$ corresponding to the $k$-th eigenvalue are uniformly bounded and converge to the set of $L^2$-normalized eigenvalues of the logarithmic Laplacian. In order to derive these spectral asymptotics, we need to establish new uniform regularity and boundary decay estimates for Dirichlet eigenfunctions for the fractional Laplacian. As a byproduct, we also obtain corresponding regularity properties of eigenfunctions of the logarithmic Laplacian.

math.AP↗
arXiv2021

Fourier-extension estimates for symmetric functions and applications to nonlinear Helmholtz equations

We establish weighted $L^p$-Fourier-extension estimates for $O(N-k) \times O(k)$-invariant functions defined on the unit sphere $\mathbb{S}^{N-1}$, allowing for exponents $p$ below the Stein-Tomas critical exponent $\frac{2(N+1)}{N-1}$. Moreover, in the more general setting of an arbitrary closed subgroup $G \subset O(N)$ and $G$-invariant functions, we study the implications of weighted Fourier-extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results for $G$-invariant solutions to the nonlinear Helmholtz equation $$ - Δu - u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}(\mathbb{R}^{N}), $$ where $Q$ is a nonnegative bounded and $G$-invariant weight function.

math.AP↗
arXiv2020

The nonlinear Schrödinger equation in the half-space

The present paper is concerned with the half-space Dirichlet problem \begin{equation} \tag{$P_c$} \label{problem-abstract} -Δv + v = |v|^{p-1}v,\ \mbox{ in } \mathbb{R}^N_{+}, \qquad v = c,\ \mbox{ on } \partial \mathbb{R}^N_{+},\ \qquad \lim_{x_N \to \infty} v(x',x_N) = 0 \mbox{ uniformly in }x' \in \mathbb{R}^{N-1}, \end{equation} where $\mathbb{R}^N_{+} := \{\,x \in \mathbb{R}^N: x_N > 0\, \}$ for some $N \geq 1$ and $p > 1$, $c > 0$ are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to \eqref{problem-abstract}. We prove that the existence and multiplicity of bounded positive solutions to \eqref{problem-abstract} depend in a striking way on the value of $c > 0$ and also on the dimension $N$. We find an explicit number $c_p \in (1,\sqrt{e})$, depending only on $p$, which determines the threshold between existence and non-existence. In particular, in dimensions $N \geq 2$, we prove that, for $0 < c < c_p$, problem \eqref{problem-abstract} admits infinitely many bounded positive solutions, whereas, for $c > c_p$, there are no bounded positive solutions to \eqref{problem-abstract}.

math.AP↗
arXiv2020

The Poisson problem for the fractional Hardy operator: Distributional identities and singular solutions

The purpose of this paper is to study and classify singular solutions of the Poisson problem $$ %\begin{equation}\label{eq 0.1} \left \{ \begin{aligned} {\mathcal L}^s_μu = f \quad\ {\rm in}\ \, Ω\setminus \{0\},\\ u =0 \quad\ {\rm in}\ \, {\mathbb R}^N \setminus Ω %\\ %\liminf_{x \to 0}\:|u(x)| /Φ_μ(x) = k. \end{aligned} \right. $$ for the fractional Hardy operator ${\mathcal L}_μ^s u= (-Δ)^s u +\fracμ{|x|^{2s}}u$ in a bounded domain $Ω\subset {\mathbb R}^N$ ($N \ge 2$) containing the origin. Here $(-Δ)^s$, $s\in(0,1)$, is the fractional Laplacian of order $2s$, and $μ\ge μ_0$, where $μ_0 = -2^{2s}\frac{Γ^2(\frac{N+2s}4)}{Γ^2(\frac{N-2s}{4})}<0$ is the best constant in the fractional Hardy inequality. The analysis requires a thorough study of fundamental solutions and associated distributional identities. Special attention will be given to the critical case $μ= μ_0$ which requires more subtle estimates than the case $μ>μ_0$.

math.AP↗
arXiv2020

Spectral properties of the logarithmic Laplacian

We obtain spectral inequalities and asymptotic formulae for the discrete spectrum of the operator $\frac12\, \log(-Δ)$ in an open set $Ω\in\Bbb R^d$, $d\ge2$, of finite measure with Dirichlet boundary conditions. We also derive some results regarding lower bounds for the eigenvalue $λ_1(Ω)$ and compare them with previously known inequalities.

math.SP↗
arXiv2020

A new look at the fractional Poisson problem via the Logarithmic Laplacian

We analyze the $s$-dependence of solutions $u_s$ to the family of fractional Poisson problems $(-Δ)^s u =f$ in $Ω$, $u \equiv 0$ on $\mathbb{R}^N\setminus Ω$ in an open bounded set $Ω\subset \mathbb{R}^N$, $s \in (0,1)$. In the case where $Ω$ is of class $C^2$ and $f \in C^α(\barΩ)$ for some $α>0$, we show that the map $(0,1) \to L^\infty(Ω)$, $s\mapsto u_s$ is of class $C^1$, and we characterize the derivative $\partial_s u_s$ in terms of the logarithmic Laplacian of $f$. As a corollary, we derive pointwise monotonicity properties of the solution map $s \mapsto u_s$ under suitable assumptions on $f$ and $Ω$. Moreover, we derive explicit bounds for the corresponding Green operator on arbitrary bounded domains which are new even for the case $s=1$, i.e., for the local Dirichlet problem $-Δu = f$ in $Ω$, $u \equiv 0$ on $\partial Ω$.

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