Search results for " 35J25"
showing 10 items of 11 documents
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.
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.
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.
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.
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.
Linearized Calder\'on problem and exponentially accurate quasimodes for analytic manifolds
2020
In this article we study the linearized anisotropic Calder\'on problem on a compact Riemannian manifold with boundary. This problem amounts to showing that products of pairs of harmonic functions of the manifold form a complete set. We assume that the manifold is transversally anisotropic and that the transversal manifold is real analytic and satisfies a geometric condition related to the geometry of pairs of intersecting geodesics. In this case, we solve the linearized anisotropic Calder\'on problem. The geometric condition does not involve the injectivity of the geodesic X-ray transform. Crucial ingredients in the proof of our result are the construction of Gaussian beam quasimodes on the…
Calder\'on's problem for p-Laplace type equations
2016
We investigate a generalization of Calder\'on's problem of recovering the conductivity coefficient in a conductivity equation from boundary measurements. As a model equation we consider the p-conductivity equation with p strictly between one and infinity, which reduces to the standard conductivity equation when p equals two, and to the p-Laplace equation when the conductivity is constant. The thesis consists of results on the direct problem, boundary determination and detecting inclusions. We formulate the equation as a variational problem also when the conductivity may be zero or infinity in large sets. As a boundary determination result we recover the first order derivative of a smooth co…
Gradient estimates for the perfect conductivity problem in anisotropic media
2018
Abstract We study the perfect conductivity problem when two perfectly conducting inclusions are closely located to each other in an anisotropic background medium. We establish optimal upper and lower gradient bounds for the solution in any dimension which characterize the singular behavior of the electric field as the distance between the inclusions goes to zero.
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.