Related papers: $H$-convergence for equations depending on monoton…
We show that for integral operators of general form the norm bounds in Lorentz spaces imply certain norm bounds for the maximal function. As a consequence, the a.e. convergence for the integral operators on the Lorentz spaces follows from…
We consider the Dirichlet problem on general, possibly nonsmooth bounded domain, for elliptic linear equation with uniformly elliptic divergence form operator. We investigate carefully the relationship between weak, soft and the…
We prove that for every planar convex set $\Omega$, the function $t\in (-r(\Omega),+\infty)\longmapsto \sqrt{|\Omega_t|}h(\Omega_t)$ is monotonically decreasing, where $r$, $|\cdot|$ and $h$ stand for the inradius, the measure and the…
The goal of this paper is to solve a long standing open problem, namely, the asymptotic development of order $2$ by $\Gamma$-convergence of the mass-constrained Cahn-Hilliard functional. This is achieved by introducing a novel rearrangement…
We present some monotonicity results for Dirichlet $L$-functions associated to real primitive characters. We show in particular that these Dirichlet $L$-functions are far from being logarithmically completely monotonic. Also, we show that,…
The aim of this paper is to establish strong convergence theorems for a strongly relatively nonexpansive sequence in a smooth and uniformly convex Banach space. Then we employ our results to approximate solutions of the zero point problem…
Monotonicity of pairs of operators is an extension of monotonicity of operators, which plays an important role in solving non-monotone inclusions. One of challenging problems in this new tool is how to design the associated mappings to…
We extend recent work of Gurel-Gurevich--Jerison--Nachmias (2020) and Bou-Rabee--Gwynne (2024) by showing that as the mesh of our lattice tends to $0$, we have a polynomial rate of convergence for the Dirichlet problem on orthodiagonal maps…
In this paper we investigate in a Hilbert space setting a second order dynamical system of the form $$\ddot{x}(t)+\g(t)\dot{x}(t)+x(t)-J_{\lambda(t) A}\big(x(t)-\lambda(t) D(x(t))-\lambda(t)\beta(t)B(x(t))\big)=0,$$ where $A:{\mathcal…
We consider the Dirichlet problem for the complex Monge-Amp\`ere equation in a bounded strongly hyperconvex Lipschitz domain in $\C^n$. We first give a sharp estimate on the modulus of continuity of the solution when the boundary data is…
In this work, we review and extend some well known results for the eigenvalues of the Dirichlet $p-$Laplace operator to a more general class of monotone quasilinear elliptic operators. As an application we obtain some homogenization results…
We study the Dirichlet problem for subelliptic partial differential equations of Monge-Ampere type involving the derivates with respect to a family X of vector fields of Carnot type. The main result is a comparison principle among viscosity…
The main result establishes the existence of a solution in a generalized sense for a nonlinear Dirichlet problem driven by a competing operator and exhibiting a convection term composed with an intrinsic operator. A finite dimensional…
We consider the Dirichlet Laplacian $\mathcal{A}_\varepsilon=-\Delta$ in the domain $\Omega\setminus\bigcup_i K_{i\varepsilon}\subset\mathbb{R}^n$ with holes $K_{i\varepsilon}$ and the Schr\"{o}dinger operator $\mathcal{A}=-\Delta+V$ in…
We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace-Beltrami operator converges to the spectrum of the (differential) Laplacian on…
Inspired by results of Bayart on ordinary Dirichlet series $\sum a_n n^{-s}$, the main purpose of this article is to start an $\mathcal{H}_p$-theory of general Dirichlet series $\sum a_n e^{-\lambda_{n}s}$. Whereas the…
We consider sequences of elliptic and parabolic operators in divergence form and depending on a family of vector fields. We show compactness results with respect to G-convergence, or H-convergence, by means of the compensated compactness…
Consider an operator equation $F(u)=0$ in a real Hilbert space. Let us call this equation ill-posed if the operator $F'(u)$ is not boundedly invertible, and well-posed otherwise. If $F$ is monotone $C^2_{loc}(H)$ operator, then we construct…
We study some Dirichlet problem for a $p$--Laplacian type operator in the setting of Orlicz--Zygmund space $L^q\log^{-\alpha}L(\Omega,\mathbb R^N)$, $q >1$ and $\alpha>0$. More precisely, our aim is to establish which assuptions on the…
First we show that the abscissae of uniform and absolute convergence of Dirichlet series coincide in the case of $L$-functions from the Selberg class $\mathcal{S}$. We also study the latter abscissa inside the extended Selberg class,…