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Jean-Baptiste Casteras

Publications and source records attributed to Jean-Baptiste Casteras.

36 records · Page 2Linked to original sources
arXiv2020

Singular radial solutions for the Lin-Ni-Takagi equation

We study singular radially symmetric solution to the Lin-Ni-Takagi equation for a supercritical power non-linearity in dimension $N\geq 3$. It is shown that for any ball and any $k \geq 0$, there is a singular solution that satisfies Neumann boundary condition and oscillates at least $k$ times around the constant equilibrium. Moreover, we show that the Morse index of the singular solution is finite or infinite if the exponent is respectively larger or smaller than the Joseph--Lundgren exponent.

math.AP↗
arXiv2019

Asymptotic Plateau problem for prescribed mean curvature hypersurfaces

We prove the existence of solutions to the asymptotic Plateau problem for hypersurfaces of prescribed mean curvature in Cartan-Hadamard manifolds $N$. More precisely, given a suitable subset $L$ of the asymptotic boundary of $N$ and a suitable function $H$ on $N$, we are able to construct a set of locally finite perimeter whose boundary has generalized mean curvature $H$ provided that $N$ satisfies the so-called strict convexity condition and that its sectional curvatures are bounded from above by a negative constant. We also obtain a multiplicity result in low dimensions.

math.DG↗
arXiv2019

Dirichlet problem for $f$-minimal graphs

We study the asymptotic Dirichlet problem for $f$-minimal graphs in Cartan-Hadamard manifolds $M$. $f$-minimal hypersurfaces are natural generalizations of self-shrinkers which play a crucial role in the study of mean curvature flow. In the first part of this paper, we prove the existence of $f$-minimal graphs with prescribed boundary behavior on a bounded domain $Ω\subset M$ under suitable assumptions on $f$ and the boundary of $Ω$. In the second part, we consider the asymptotic Dirichlet problem. Provided that $f$ decays fast enough, we construct solutions to the problem. Our assumption on the decay of $f$ is linked with the sectional curvatures of $M$. In view of a result of Pigola, Rigoli and Setti, our results are almost sharp.

math.DG↗
arXiv2019

Asymptotic Dirichlet problems in warped products

We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature $H$ in warped product manifolds $M\times_\varrho \mathbb{R}$. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on $H$ and the mean curvature of the Killing cylinders over geodesic spheres. In the process we obtain a uniform interior gradient estimate improving previous results by Dajczer and de Lira. In the second part we solve the asymptotic Dirichlet problem in a large class of manifolds whose sectional curvatures are allowed to go to $0$ or to $-\infty$ provided that $H$satisfies certain bounds with respect to the sectional curvatures of $M$ and the norm of the Killing vector field. Finally we obtain non-existence results if the prescribed mean curvature function $H$ grows too fast.

math.DG↗
arXiv2019

Existence and non-existence of minimal graphic and $p$-harmonic functions

We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold $M$ with only one end if $M$ has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constant minimal graphic and $p$-harmonic functions on rotationally symmetric Cartan-Hadamard manifolds under optimal assumptions on the sectional curvatures.

math.DG↗
arXiv2019

Construction of a solution for the two-component radial Gross-Pitaevskii system with a large coupling parameter

We consider strongly coupled competitive elliptic systems that arise in the study of two-component Bose-Einstein condensates. As the coupling parameter tends to infinity, solutions that remain uniformly bounded are known to converge to a segregated limiting profile, with the difference of its components satisfying a limit scalar PDE. In the case of radial symmetry, under natural non-degeneracy assumptions on a solution of the limit problem, we establish by a perturbation argument its persistence as a solution to the elliptic system.

math.AP↗
arXiv2019

On Helmholtz equations and counterexamples to Strichartz estimates in hyperbolic space

In this paper, we study nonlinear Helmholtz equations (NLH) $-Δ_{\mathbb{H}^N} u - \frac{(N-1)^2}{4} u -λ^2 u = Γ|u|^{p-2}u$ in $\mathbb{H}^N$, $N\geq 2$ where $Δ_{\mathbb{H}^N}$ denotes the Laplace-Beltrami operator in the hyperbolic space $\mathbb{H}^N$ and $Γ\in L^\infty(\mathbb{H}^N)$ is chosen suitably. Using fixed point and variational techniques, we find nontrivial solutions to (NLH) for all $λ>0$ and $p>2$. The oscillatory behaviour and decay rates of radial solutions is analyzed, with possible extensions to Cartan-Hadamard manifolds and Damek-Ricci spaces. Our results rely on a new Limiting Absorption Principle for the Helmholtz operator in $\mathbb{H}^N$. As a byproduct, we obtain simple counterexamples to certain Strichartz estimates.

math.AP↗
arXiv2018

Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equation $$ γΔ^2 u -Δu + αu=|u|^{2 σ} u, \quad u \in H^2(\R^N), $$ under the constraint $$ \int_{\R^N}|u|^2 \, dx =c>0. $$ We assume $γ>0, N \geq 1, 4 \leq σN < \frac{4N}{(N-4)^+}$, whereas the parameter $α\in \R$ will appear as a Lagrange multiplier. Given $c \in \R^+$, we consider several questions including the existence of ground states, of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation.

math.AP↗
arXiv2018

On the asymptotic Dirichlet problem for a class of mean curvature type partial differential equations

We study the Dirichlet problem for the following prescribed mean curvature PDE $$ \begin{cases} -\operatorname{div}\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}=f(x,v) \text{ in }Ω\\ v=φ\text{ on }\partialΩ. \end{cases} $$ where $Ω$ is a domain contained in a complete Riemannian manifold $M,$ $f:Ω\times\mathbb{R\rightarrow R}$ is a fixed function and $φ$ is a given continuous function on $\partialΩ$. This is done in three parts. In the first one we consider this problem in the most general form, proving the existence of solutions when $Ω$ is a bounded $C^{2,α}$ domain, under suitable conditions on $f$, with no restrictions on $M$ besides completeness. In the second part we study the asymptotic Dirichlet problem when $M$ is the hyperbolic space $\mathbb{H}^n$ and $Ω$ is the whole space. This part uses in an essential way the geometric structure of $\mathbb{H}^n$ to construct special barriers which resemble the Scherk type solutions of the minimal surface PDE. In the third part one uses these Scherk type graphs to prove the non existence of isolated asymptotic boundary singularities for global solutions of this Dirichlet problem.

math.DG↗
arXiv2018

Singular radial solutions for the Keller-Segel equation in high dimension

We study singular radially symmetric solution of the stationary Keller-Segel equation, that is, an elliptic equation with exponential nonlinearity, which is super-critical in dimension $N \geq 3$. The solutions are unbounded at the origin and we show that they describe the asymptotics of bifurcation branches of regular solutions. It is shown that for any ball and any $k \geq 0$, there is a singular solution that satisfies Neumann boundary condition and oscillates at least $k$ times around the constant equilibrium. Moreover, we prove that in dimension $3 \leq N \leq 9$ there are regular solutions satisfying Neumann boundary conditions that are close to singular ones. Hence, it follows that there exist regular solutions on any ball with arbitrarily fast oscillations. For generic radii, we show that the bifurcation branches of regular solutions oscillate in the bifurcation plane when $3\leq N\leq 9$ and approach to a singular solution. In dimension $N > 10$, we show that the Morse index of the singular solution is finite, and therefore the existence of regular solutions with fast oscillations is not expected.

math.AP↗
arXiv2018

On a fourth order nonlinear Helmholtz equation

In this paper, we study the mixed dispersion fourth order nonlinear Helmholtz equation $Δ^2 u -βΔu + αu= Γ|u|^{p-2} u$ in $\mathbb R^N$ for positive, bounded and $\mathbb Z^N$-periodic functions $Γ$. Using the dual method of Evequoz and Weth, we find solutions to this equation and establish some of their qualitative properties.

math.AP↗
arXiv2017

Strong instability of ground states to a fourth order Schrödinger equation

In this note we prove the instability by blow-up of the ground state solutions for a class of fourth order Schr\" odinger equations. This extends the first rigorous results on blowing-up solutions for the biharmonic NLS due to Boulenger and Lenzmann \cite{BoLe} and confirm numerical conjectures from \cite{BaFi, BaFiMa1, BaFiMa, FiIlPa}.

math.AP↗
arXiv2016

Solvability of minimal graph equation under pointwise pinching condition for sectional curvatures

We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan-Hadamard manifold $M$ whose radial sectional curvatures outside a compact set satisfy an upper bound $$K(P)\le - \frac{ϕ(ϕ-1)}{r(x)^2}$$ and a pointwise pinching condition $$|K(P)|\le C_K|K(P')|$$ for some constants $ϕ>1$ and $C_K\ge 1$, where $P$ and $P'$ are any 2-dimensional subspaces of $T_xM$ containing the (radial) vector $\nabla r(x)$ and $r(x)=d(o,x)$ is the distance to a fixed point $o\in M$. We solve the asymptotic Dirichlet problem with any continuous boundary data for dimensions $n>4/ϕ+1$.

math.DG↗
arXiv2016

Multiple positive solutions of the stationary Keller-Segel system

We consider the stationary Keller-Segel equation \begin{equation*} \begin{cases} -Δv+v=λe^v, \quad v>0 \quad & \text{in }Ω,\\ \partial_νv=0 &\text{on } \partial Ω, \end{cases} \end{equation*} where $Ω$ is a ball. In the regime $λ\to 0$, we study the radial bifurcations and we construct radial solutions by a gluing variational method. For any given natural positive number $n$, we build a solution having multiple layers at $r_1,\ldots,r_n$ by which we mean that the solutions concentrate on the spheres of radii $r_i$ as $λ\to 0$ (for all $i=1,\ldots,n$). A remarkable fact is that, in opposition to previous known results, the layers of the solutions do not accumulate to the boundary of $Ω$ as $λ\to 0$. Instead they satisfy an optimal partition problem in the limit.

math.AP↗
arXiv2014

On the asymptotic Plateau's problem for CMC hypersurfaces on rank 1 symmetric spaces of noncompact type

Let $M$ be a Hadamard manifold with curvature bounded above by a negative constant $-α$, satisfying the "strict convexity condition", and assume that $M$ admits a "helicoidal" one-parameter subgroup $G$ of isometries of $M$. Then, given a compact topological $G-$shaped hypersurface $Γ$ in the asymptotic boundary of $M,$ and $|H|<\sqrtα$, we prove the existence of a complete properly embedded hypersurface whose mean curvature is equal to $H$ and whose asymptotic boundary is $Γ$. We are able, this way, to extend a previous theorem of B.Guan and J.Spruck on the hyperbolic space to any rank 1 symmetric spaces of non compact type and to more general boundary data.

math.DG↗
arXiv2012

Quantitative uniqueness for Schrodinger operator with regular potentials

We give a sharp upper bound on the vanishing order of solutions to Schrodinger equation with C^1 electric and magnetic potentials on a compact smooth manifold. Our method is based on quantitative Carleman type inequalities developed by Donnelly and Fefferman. It also extends the first author's previous work to the magnetic potential case.

math.AP↗