This note provides sharp regularity results for general, time-independent, second order, hyperbolic equations with non-homogeneous data of Neumann type.

We describe the asymptotic distribution of eigenvalues of self-adjoint elliptic differential operators, assuming that the first-order derivatives of the coefficients are Lipschitz continuous. We consider the asymptotic formula of Hörmander's type for the spectral function of pseudodifferential operators obtained via a regularization procedure of non-smooth coefficients.

In this paper we prove a two-term asymptotic formula for the spectral counting function for a $2$D magnetic Schrödinger operator on a domain (with Dirichlet boundary conditions) in a semiclassical limit and with strong magnetic field. By scaling, this is equivalent to a thermodynamic limit of a $2$D Fermi gas submitted to a constant external magnetic field.The original motivation comes from a paper by H. Kunz in which he studied, among other things, the boundary correction for the grand-canonical pressure...

We construct an upper bound for the following family of functionals ${\left\{E_{\epsilon }\right\}}_{\epsilon \>0}$, which arises in the study of micromagnetics:$$E_{\epsilon }\left(u\right)={\int }_{\Omega }{\epsilon \left|\nabla u\right|}^2+\frac{1}{\epsilon }{\int }_{{ℝ}^2}{\left|H_u\right|}^2.$$Here $\Omega$ is a bounded domain in ${ℝ}^2$, $u\in H^1(\Omega ,S^1)$ (corresponding to the magnetization) and $H_u$, the demagnetizing field created by $u$, is given by$$\left\{\begin{array}{cc}\mathrm{div}(\tilde{u}+H_u)=0\hfill & \text{in}{ℝ}^2,\hfill \\ \mathrm{curl}{H}_u=0\hfill & \text{in}{ℝ}^2,\hfill \end{array}\right.$$where $\tilde{u}$ is the extension of $u$ by $0$ in ${ℝ}^2\setminus \Omega$. Our upper bound coincides with the lower bound obtained by Rivière and Serfaty.

A review of some methods in sheaf theory is presented to make precise a general concept of regularity in algebras or spaces of generalized functions. This leads to the local analysis of the sections of sheaves or presheaves under consideration and then to microlocal analysis and microlocal asymptotic analysis.

We consider the flow of a class of incompressible fluids which are constitutively defined by the symmetric part of the velocity gradient being a function, which can be non-monotone, of the deviator of the stress tensor. These models are generalizations of the stress power-law models introduced and studied by J. Málek, V. Průša, K. R. Rajagopal: Generalizations of the Navier-Stokes fluid from a new perspective. Int. J. Eng. Sci. 48 (2010), 1907–1924. We discuss a potential application of the new...

We prove a characterisation of sets with finite perimeter and $BV$ functions in terms of the short time behaviour of the heat semigroup in ${\mathbf{R}}^N$. For sets with smooth boundary a more precise result is shown.

We prove the existence of positive and of nodal solutions for $-\Delta u={\left|u\right|}^{p-2}u+\mu {\left|u\right|}^{q-2}u$, $u\in {\mathrm{H}}_0^1\left(\Omega \right)$, where $\mu \>0$ and $2\<q\<p=2N(N-2)$, for a class of open subsets $\Omega$ of ${ℝ}^N$ lying between two infinite cylinders.