On convolution type functional equations.
Discrete spectral synthesis.
Functional equations on Sturm-Liouville hypergroups.
D'Alembert's functional equation on compact groups.
On a linear functional equation.
Mean-value type functional equations.
Fréchet's equation and Hyers theorem on noncommutative semigroups
Remark on a paper of M. A. McKiernan
On quasiconvex hulls in symmetric matrices
Superstability of functional equations related to spherical functions
In this paper we prove stability-type theorems for functional equations related to spherical functions. Our proofs are based on superstability-type methods and on the method of invariant means.
The Levi-Civita equation, vector modules and spectral synthesis
The purpose of this paper is to give a survey on some recent results concerning spectral analysis and spectral synthesis in the framework of vector modules and in close connection with the Levi-Civita functional equation. Further, we present some open problems in this subject.
On the extension of exponential polynomials
Exponential polynomials are the building bricks of spectral synthesis. In some cases it happens that exponential polynomials should be extended from subgroups to whole groups. To achieve this aim we prove an extension theorem for exponential polynomials which is based on a classical theorem on the extension of homomorphisms.
Relaxation of the incompressible porous media equation
It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding configurations. We then use this to construct weak solutions to the unstable interface problem (the...
Dissipative Euler flows and Onsager's conjecture
Building upon the techniques introduced in [15], for any we construct periodic weak solutions of the incompressible Euler equations which dissipate the total kinetic energy and are Hölder-continuous with exponent . A famous conjecture of Onsager states the existence of such dissipative solutions with any Hölder exponent . Our theorem is the first result in this direction.
Convex integration and the theory of elliptic equations
This paper deals with the theory of linear elliptic partial differential equations with bounded measurable coefficients. We construct in two dimensions examples of weak and so-called very weak solutions, with critical integrability properties, both to isotropic equations and to equations in non-divergence form. These examples show that the general theory, developed in [1, 24] and [2], cannot be extended under any restriction on the essential range of the coefficients. Our constructions are based...
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