Related papers: Radial symmetry and partially overdetermined probl…
We formulate on rectangles and on the right horizontal half-strip initial-boundary value problems for a two-dimensional Benney-Lin type equation. Existence and uniqueness of a regular solution as well as the exponential decay rate for the…
The convex shape contained in a disk having prescribed area and maximal perimeter is completely characterized in terms of the area fraction. The solution is always a polygon having all but one sides equal. The lengths of the sides are…
Second order nonlinear eigenvalue problems are considered for which the spectrum is an interval. The boundary conditions are of Robin and Dirichlet type. The shape and the number of solutions are discussed by means of a phase plane…
Overdetermined systems of first kind integral equations appear in many applications. When the right-hand side is discretized, the resulting finite-data problem is ill-posed and admits infinitely many solutions. We propose a numerical method…
The existence and multiplicity and nonexistence of nontrivial radial convex solutions of systems of Monge-Amp\`ere equations are established with superlinearity or sublinearity assumptions for an appropriately chosen parameter. The proof of…
The affine inverse eigenvalue problem consists of identifying a real symmetric matrix with a prescribed set of eigenvalues in an affine space. Due to its ubiquity in applications, various instances of the problem have been widely studied in…
Consider a polyhedral convex cone which is given by a finite number of linear inequalities. We investigate the problem to project this cone into a subspace and show that this problem is closely related to linear vector optimization: We…
This paper deals with existence and multiplicity of positive solutions for a quasilinear problem with Neumann boundary conditions, set in a ball. The problem admits at least one constant non-zero solution and it involves a nonlinearity that…
In this paper, we prove the symmetry of the solution to overdetermined problem for the equation $\sigma_k(D^2u-uI)=C_n^k$ in hyperbolic space. Our approach is based on establishing a Rellich-Pohozaev type identity and using a P function.…
We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which are convex combinations of a small number of atoms. These atoms are identified with the extreme points and…
We shall prove a multiplicity result for semilinear elliptic problems with a super-critical nonlinearity of the form, \begin{equation}\label{con-c} \left \{ \begin{array}{ll} -\Delta u =|u|^{p-2} u+\mu |u|^{q-2}u, & x \in \Omega\\ u=0, & x…
Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its…
In this paper, we investigate the existence of positive solutions for a nonlocal fractional boundary value problem involving Caputo fractional derivative and nonlocal Riemann-Stieltjes integral boundary condition. By using the spectral…
This article investigates the numerical approximation of shape optimization problems with PDE constraint on classes of convex domains. The convexity constraint provides a compactness property which implies well posedness of the problem.…
We study the problem of super-resolution of a linear combination of Dirac distributions and their derivatives on a one-dimensional circle from noisy Fourier measurements. Following numerous recent works on the subject, we consider the…
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.…
We show that there is a query expressible in first-order logic over the reals that returns, on any given semi-algebraic set A, for every point a radius around which A is conical. We obtain this result by combining famous results from…
We consider radially symmetric solutions for a class of resonant problems on a unit ball $B \subset R^n$ around the origin \[ \Delta u+\la _1 u +g(u)=f(r) \s \mbox{for $x \in B$}, \s u=0 \s \mbox{on $\partial B$} \,. \] Here the function…
We consider an overdetermined Serrin's type problem in space forms and we generalize Weinberger's proof in [Arch. Rational Mech. Anal., 43 (1971)] by introducing a suitable P-function.
We consider the Dirichlet problem for a class of semilinear equations on two dimensional convex domains. We give a sufficient condition for the solution to be concave. Our condition uses comparison with ellipses, and is motivated by an idea…