arXiv subjects

Tobias Weth

Publications and source records attributed to Tobias Weth.

At least 73 records · Page 4Linked to original sources

arXiv2015

Dual variational methods and nonvanishing for the nonlinear Helmholtz equation

We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation $$ -Δu-k^2 u =Q(x)|u|^{p-2}u,\qquad u \in W^{2,p}(\mathbb{R}^N) $$ with $N\geq 3$, $\frac{2(N+1)}{(N-1)}< p<\frac{2N}{N-2}$ and nonnegative $Q \in L^\infty(\mathbb{R}^N)$. We prove the existence of nontrivial solutions for periodic $Q$ as well as in the case where $Q(x)\to 0$ as $|x|\to\infty$. In the periodic case, a key ingredient of the approach is a new nonvanishing theorem related to an associated integral equation. The solutions we study are superpositions of outgoing and incoming waves and are characterized by a nonlinear far field relation.

math.AP↗

arXiv2015

Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay

We are concerned with hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in $\mathbb{R}^N$ with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or "cylinders" in $\mathbb{R}^2$ with constant nonlocal mean curvature and bifurcating from a straight band. These are Delaunay type bands in the nonlocal setting. Here we use a Lyapunov-Schmidt procedure for a quasilinear type fractional elliptic equation.

math.AP↗

arXiv2015

Existence, unique continuation and symmetry of least energy nodal solutions to sublinear Neumann problems

We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c l c} -Δu & = &|u|^{q-2}u & \textrm{in }Ω, \\ u_n & = & 0 & \textrm{on }\partialΩ,\end{array}\right. \end {equation*} where $Ω\subset \real^N$ is a bounded domain, and $1 \leq q < 2$. For $q=1$, $|u|^{q-2}u$ will be identified with $\sgn(u)$. We establish a variational principle for least energy nodal solutions, and we investigate their qualitative properties. In particular, we show that they satisfy a unique continuation property (their zero set is Lebesgue-negligible). Moreover, if $Ω$ is radial, then least energy nodal solutions are foliated Schwarz symmetric, and they are nonradial in case $Ω$ is a ball. The case $q=1$ requires special treatment since the formally associated energy functional is not differentiable, and many arguments have to be adjusted.

math.AP↗

arXiv2015

Profile expansion for the first nontrivial Steklov eigenvalue in Riemannian manifolds

We study the problem of maximizing the first nontrivial Steklov eigenvalue of the Laplace-Beltrami Operator among subdomains of fixed volume of a Riemannian manifold. More precisely, we study the expansion of the corresponding profile of this isoperimetric (or isochoric) problem as the volume tends to zero. The main difficulty encountered in our study is the lack of existence results for maximizing domains and the possible degeneracy of the first nontrivial Steklov eigenvalue, which makes it difficult to tackle the problem with domain variation techniques. As a corollary of our results, we deduce local comparison principles for the profile in terms of the scalar curvature on $\mathcal{M}$. In the case where the underlying manifold is a closed surface, we obtain a global expansion and thus a global comparison principle.

math.AP↗

arXiv2015

On the solvability of relaxed one-sided Lipschitz inclusions in Hilbert spaces

We prove solvability theorems for relaxed one-sided Lipschitz multivalued mappings in Hilbert spaces and for composed mappings in the Gelfand triple setting. From these theorems, we deduce properties of the inverses of such mappings and convergence properties of a numerical scheme for the solution of algebraic inclusions.

math.OC↗

arXiv2014

On the asymptotic shape of solutions to Neumann problems for non-cooperative parabolic systems

We consider a class of nonautonomous parabolic competition-diffusion systems on bounded radial domains under Neumann boundary conditions. We show that, if the initial profiles satisfy a reflection inequality with respect to a hyperplane, then global positive solutions are asymptotically (in time) foliated Schwarz symmetric with respect to antipodal points. Additionally, a related result for (positive and sign changing solutions) of a scalar equation with Neumann boundary conditions is given. The asymptotic shape of solutions to cooperative systems is also discussed.

math.AP↗

arXiv2014

The shape of extremal functions for Poincaré-Sobolev-type inequalities in a ball

We study extremal functions for a family of Poincaré-Sobolev-type inequalities. These functions minimize, for subcritical or critical $p\geq 2$, the quotient ${\|\nabla u\|_2}/{\|u\|_p}$ among all $u \in H^1(B)\setminus\{0\}$ with $\int_{B}u=0$. Here $B$ is the unit ball in $\mathbb{R}^N$. We show that the minimizers are axially symmetric with respect to a line passing through the origin. We also show that they are strictly monotone in the direction of this line. In particular, they take their maximum and minimum precisely at two antipodal points on the boundary of $B$. We also prove that, for $p$ close to $2$, minimizers are antisymmetric with respect to the hyperplane through the origin perpendicular to the symmetry axis, and that, once the symmetry axis is fixed, they are unique (up to multiplication by a constant). In space dimension two, we prove that minimizers are not antisymmetric for large $p$.

math.AP↗

arXiv2014

Symmetry via antisymmetric maximum principles in nonlocal problems of variable order

We consider the nonlinear problem \[(P) \;\; I u=f(x,u) \text{ in $Ω$,} \;\; u=0 \text{ on $\mathbb{R}^{N}\setminusΩ$ }\] in an open bounded set $Ω\subset\mathbb{R}^{N}$, where $I$ is a nonlocal operator which may be anisotropic and may have varying order. We assume mild symmetry and monotonicity assumptions on $I$, $Ω$ and the nonlinearity $f$ with respect to a fixed direction, say $x_1$, and we show that any nonnegative weak solution $u$ of $(P)$ is symmetric in $x_1$. Moreover, we have the following alternative: Either $u\equiv 0$ in $Ω$, or $u$ is strictly decreasing in $|x_1|$. The proof relies on new maximum principles for antisymmetric supersolutions of an associated class of linear problems.

math.AP↗

arXiv2013

Monotonicity and nonexistence results for some fractional elliptic problems in the half space

We study a class of fractional elliptic problems of the form $\Ds u= f(u)$ in the half space $\R^N_+:=\{x \in \R^N\::\: x_1>0\}$ with the complementary Dirichlet condition $u \equiv 0$ in $\R^N \setminus \R^N_+$. Under mild assumptions on the nonlinearity $f$, we show that bounded positive solutions are increasing in $x_1$. For the special case $f(u)=u^q$, we deduce nonexistence of positive bounded solutions in the case where $q \ge 1$ and $q<\frac{N-1+2s}{N-1-2s}$ if $N \ge 1+2s$. We do not require integrability assumptions on the solutions we study.

math.AP↗

arXiv2013

Sharp local estimates for the Szegö-Weinberger profile in Riemannian manifolds

We study the local Szegö-Weinberger profile in a geodesic ball $B_g(y_0,r_0)$ centered at a point $y_0$ in a Riemannian manifold $(\M,g)$. This profile is obtained by maximizing the first nontrivial Neumann eigenvalue $μ_2$ of the Laplace-Beltrami Operator $Δ_g$ on $\M$ among subdomains of $B_g(y_0,r_0)$ with fixed volume. We derive a sharp asymptotic bounds of this profile in terms of the scalar curvature of $\M$ at $y_0$. As a corollary, we deduce a local comparison principle depending only on the scalar curvature. Our study is related to previous results on the profile corresponding to the minimization of the first Dirichlet eigenvalue of $Δ_g$, but additional difficulties arise due to the fact that $μ_2$ is degenerate in the unit ball in $\R^N$ and geodesic balls do not yield the optimal lower bound in the asymptotics we obtain.

math.DG↗

arXiv2013

Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations

We study the nonlinear fractional reaction diffusion equation $\partial_{t}u + (-Δ)^{s} u= f(t,x,u)$, $s\in(0,1)$ in a bounded domain $Ω$ together with Dirichlet boundary conditions on $\R^N \setminus Ω$. We prove asymptotic symmetry of nonnegative globally bounded solutions in the case where the underlying data obeys some symmetry and monotonicity assumptions. More precisely, we assume that $Ω$ is symmetric with respect to reflection at a hyperplane, say ${x_1=0}$, and convex in the $x_1$-direction, and that the nonlinearity $f$ is even in $x_1$ and nonincreasing in $|x_1|$. Under rather weak additional technical assumptions, we then show that any nonzero element in the $ω$-limit set of nonnegative globally bounded solution is even in $x_1$ and strictly decreasing in $|x_1|$. This result, which is obtained via a series of new estimates for antisymmetric supersolutions of a corresponding family of linear equations, implies a strong maximum type principle which is not available in the non-fractional case $s=1$.

math.AP↗

arXiv2013

Real solutions to the nonlinear Helmholtz equation with local nonlinearity

In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - Δu - k^2 u = f(x,u),\qquad x\in \R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad \text{and} \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2 u(x)) =o(|x|^{\frac{1-N}{2}}) \qquad \text{as $r=|x| \to \infty$.} $$ We develop the variational framework to prove the existence of nontrivial solutions for compactly supported nonlinearities without any symmetry assumptions. In addition, we consider the radial case in which, for a larger class of nonlinearities, infinitely many solutions are shown to exist. Our results give rise to the existence of standing wave solutions of corresponding nonlinear Klein-Gordon equations with arbitrarily large frequency.

math.AP↗

arXiv2012

Nonexistence results for a class of fractional elliptic boundary value problems

In this paper we study a class of fractional elliptic problems of the form $$ \Ds u= f(x,u) \quad \textrm{in} Øu=0\quad \textrm{in} \R^N \setminus Ø,$$ where $s\in(0,1)$. We prove nonexistence of positive solutions when $Ø$ is star-shaped and $f$ is supercritical. We also derive a nonexistence result for subcritical $f$ in some unbounded domains. The argument relies on the method of moving spheres applied to a reformulated problem using the Caffarelli-Silvestre extension \cite{CSilv} of a solution of the above problem. The standard approach in the case $s=1$ using Pohozaev type identities does not carry over to the case $0<s<1$ due to the lack of boundary regularity of solutions.

math.AP↗

arXiv2012

Remainder terms in the fractional Sobolev inequality

We show that the fractional Sobolev inequality for the embedding $\H \hookrightarrow L^{\frac{2N}{N-s}}(\R^N)$, $s \in (0,N)$ can be sharpened by adding a remainder term proportional to the distance to the set of optimizers. As a corollary, we derive the existence of a remainder term in the weak $L^{\frac{N}{N-s}}$-norm for functions supported in a domain of finite measure. Our results generalize earlier work for the non-fractional case where $s$ is an even integer.

math.AP↗

arXiv2012

Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains

We consider solutions of some nonlinear parabolic boundary value problems in radial bounded domains whose initial profile satisfy a reflection inequality with respect to a hyperplane containing the origin. We show that, under rather general assumptions, these 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 nonincreasing in the polar angle from this axis. In this form, the result is new even for equilibria (i.e. solutions of the corresponding elliptic problem) and time periodic solutions.

math.AP↗

arXiv2012

Existence and symmetry results for competing variational systems

In this paper we consider a class of gradient systems of type $$ -c_i Δu_i + V_i(x)u_i=P_{u_i}(u),\quad u_1,..., u_k>0 \text{in}Ω, \qquad u_1=...=u_k=0 \text{on} \partial Ω, $$ in a bounded domain $Ω\subseteq \R^N$. Under suitable assumptions on $V_i$ and $P$, we prove the existence of ground-state solutions for this problem. Moreover, for $k=2$, assuming that the domain $Ω$ and the potentials $V_i$ are radially symmetric, we prove that the ground state solutions are foliated Schwarz symmetric with respect to antipodal points. We provide several examples for our abstract framework.

math.AP↗

arXiv2011

Entire solutions to nonlinear scalar field equations with indefinite linear part

We consider the stationary semilinear Schrödinger equation $-Δu + a(x) u = f(x,u)$, $u\in H^1(\R^N)$, where $a$ and $f$ are continuous functions converging to some limits $a_\infty>0$ and $f_\infty=f_\infty(u)$ as $|x|\to\infty$. In the indefinite setting where the Schrödinger operator $-Δ+a$ has negative eigenvalues, we combine a reduction method with a topological argument to prove the existence of a solution of our problem under weak one-sided asymptotic estimates. The minimal energy level need not be attained in this case. In a second part of the paper, we prove the existence of ground-state solutions under more restrictive assumptions on $a$ and $f$. We stress that for some of our results we also allow zero to lie in the spectrum of $-Δ+ a$.

math.AP↗

arXiv2011

Increasing radial solutions for Neumann problems without growth restrictions

We study the existence of positive increasing radial solutions for superlinear Neumann problems in the ball. We do not impose any growth condition on the nonlinearity at infinity and our assumptions allow for interactions with the spectrum. In our approach we use both topological and variational arguments, and we overcome the lack of compactness by considering the cone of nonnegative, nondecreasing radial functions of H^1.

math.AP↗