Search results for "35j25"
showing 10 items of 22 documents
Inverse problems for elliptic equations with fractional power type nonlinearities
2020
We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. By using a fractional order adaptation of this method, we show that the results of [LLLS20a, LLLS20b] remain valid for general power type nonlinearities.
The Linearized Calderón Problem in Transversally Anisotropic Geometries
2017
In this article we study the linearized anisotropic Calderon problem. In a compact manifold with boundary, this problem amounts to showing that products of harmonic functions form a complete set. Assuming that the manifold is transversally anisotropic, we show that the boundary measurements determine an FBI type transform at certain points in the transversal manifold. This leads to recovery of transversal singularities in the linearized problem. The method requires a geometric condition on the transversal manifold related to pairs of intersecting geodesics, but it does not involve the geodesic X-ray transform which has limited earlier results on this problem.
Multiplicity of positive solutions for a degenerate nonlocal problem with p-Laplacian
2021
Abstract We consider a nonlinear boundary value problem with degenerate nonlocal term depending on the L q -norm of the solution and the p-Laplace operator. We prove the multiplicity of positive solutions for the problem, where the number of solutions doubles the number of “positive bumps” of the degenerate term. The solutions are also ordered according to their L q -norms.
A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient
2019
Let $\Omega \subset \mathbb R^N$, $N \geq 2$, be a smooth bounded domain. We consider a boundary value problem of the form $$-\Delta u = c_{\lambda}(x) u + \mu(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(\Omega)\cap L^{\infty}(\Omega)$$ where $c_{\lambda}$ depends on a parameter $\lambda \in \mathbb R$, the coefficients $c_{\lambda}$ and $h$ belong to $L^q(\Omega)$ with $q>N/2$ and $\mu \in L^{\infty}(\Omega)$. Under suitable assumptions, but without imposing a sign condition on any of these coefficients, we obtain an a priori upper bound on the solutions. Our proof relies on a new boundary weak Harnack inequality. This inequality, which is of independent interest, is established in the gener…
On a generalisation of Krein's example
2017
We generalise a classical example given by Krein in 1953. We compute the difference of the resolvents and the difference of the spectral projections explicitly. We further give a full description of the unitary invariants, i.e., of the spectrum and the multiplicity. Moreover, we observe a link between the difference of the spectral projections and Hankel operators.
The Poisson embedding approach to the Calderón problem
2020
We introduce a new approach to the anisotropic Calder\'on problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calder\'on type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result by Lassas and Uhlmann (2001) solving the Calder\'on problem on real analytic Riemannian manifolds. The proof uses the Poisson embedding to determine the harmonic functions in the manifold up to a harmonic morphism. The method also involves various Runge approximation results for linear elliptic equations.
Multi-parameter analysis of the obstacle scattering problem
2022
Abstract We consider the acoustic field scattered by a bounded impenetrable obstacle and we study its dependence upon a certain set of parameters. As usual, the problem is modeled by an exterior Dirichlet problem for the Helmholtz equation Δu + k 2 u = 0. We show that the solution u and its far field pattern u ∞ depend real analytically on the shape of the obstacle, the wave number k, and the Dirichlet datum. We also prove a similar result for the corresponding Dirichlet-to-Neumann map.
Global representation and multiscale expansion for the Dirichlet problem in a domain with a small hole close to the boundary
2019
For each pair (Formula presented.) of positive parameters, we define a perforated domain (Formula presented.) by making a small hole of size (Formula presented.) in an open regular subset (Formula presented.) of (Formula presented.) ((Formula presented.)). The hole is situated at distance (Formula presented.) from the outer boundary (Formula presented.) of the domain. Thus, when (Formula presented.) both the size of the hole and its distance from (Formula presented.) tend to zero, but the size shrinks faster than the distance. Next, we consider a Dirichlet problem for the Laplace equation in the perforated domain (Formula presented.) and we denote its solution by (Formula presented.) Our ai…
An inverse problem for the minimal surface equation
2022
We use the method of higher order linearization to study an inverse boundary value problem for the minimal surface equation on a Riemannian manifold $(\mathbb{R}^n,g)$, where the metric $g$ is conformally Euclidean. In particular we show that with the knowledge of Dirichlet-to-Neumann map associated to the minimal surface equation, one can determine the Taylor series of the conformal factor $c(x)$ at $x_n=0$ up to a multiplicative constant. We show this both in the full data case and in some partial data cases.
Existence of two solutions for singular Φ-Laplacian problems
2022
AbstractExistence of two solutions to a parametric singular quasi-linear elliptic problem is proved. The equation is driven by theΦ\Phi-Laplacian operator, and the reaction term can be nonmonotone. The main tools employed are the local minimum theorem and the Mountain pass theorem, together with the truncation technique. GlobalC1,τ{C}^{1,\tau }regularity of solutions is also investigated, chiefly viaa prioriestimates and perturbation techniques.