Related papers: The Dirichlet Problem for the Logarithmic Laplacia…

It is well known that, with a particular choice of norm, the classical double-layer potential operator $D$ has essential norm $<1/2$ as an operator on the natural trace space $H^{1/2}(\Gamma)$ whenever $\Gamma$ is the boundary of a bounded…

We obtain the inequalities of the form $$\int_{\Omega}|\nabla u(x)|^2h(u(x))\,{\rm d} x\leq C\int_{\Omega} \left( \sqrt{ |P u(x)||{\cal T}_{H}(u(x))|}\right)^{2}h(u(x))\,{\rm d} x +\Theta,$$ where $\Omega\subset \mathbf{R}^n$ is a bounded…

Let $\mathcal{L}$ be a second-order linear elliptic operator with complex coefficients. We show that if the $L^p$ Dirichlet problem for the elliptic system $\mathcal{L}(u)=0$ in a fixed Lipschitz domain $\Omega$ in $\mathbb{R}^d$ is…

On a bounded domain $\Omega\subset\mathbb R^{n+1}$, $n\geq2$, satisfying the corkscrew condition and with Ahlfors regular boundary, we characterize the dual space to the space ${\bf N}_{2,p}$ of functions $u$ whose Kenig-Pipher modified…

In this paper we introduce two new fractional versions of the Laplacian. The first one is based on the classical formula that writes the usual Laplacian as the sum of the eigenvalues of the Hessian. The second one comes from looking at the…

The $p$-Laplacian operator $\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$ is not uniformly elliptic for any $p\in(1,2)\cup(2,\infty)$ and degenerates even more when $p\to \infty$ or $p\to 1$. In those two cases the Dirichlet…

Assuming the Generalized Riemann Hypothesis, we establish explicit bounds in the $q$-aspect for the logarithmic derivative $\left(L'/L\right)\left(\sigma,\chi\right)$ of Dirichlet $L$-functions, where $\chi$ is a primitive character modulo…

The purpose of this brief paper is to prove De Giorgi type results for stable solutions of the following nonlocal system of integral equations in two dimensions $$ L(u_i) = H_i(u) \quad \text{in} \ \ \mathbb R^2 , $$ where $u=(u_i)_{i=1}^m$…

In this paper we are concerned with a general singular Dirichlet boundary value problem whose model is the following $$ \begin{cases} -\Delta u = \frac{\mu}{u^{\gamma}} & \text{in}\ \Omega, u=0 &\text{on}\ \partial\Omega, u>0 &\text{on}\…

The present paper pioneers the study of the Dirichlet problem with $L^q$ boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in $\Omega := \mathbb R^n \setminus \mathbb R^d$…

Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ with smooth non-empty boundary $\partial M$. Let $\Gamma\subset\mathbb{R}^n$ be a symmetric convex cone and $f$ a symmetric defining function for $\Gamma$…

We establish a nonlinear Calder\'on-Zygmund $L^2$-theory to the Dirichlet problem $$-|Du|^{\gamma}\Delta^N_p u=f\in L^2(\Omega)\quad {\rm in}\quad \Omega; \quad u=0 \ \mbox{on $\partial\Omega$} $$ for $n\ge2$, $ p>1$ and a large range of…

In this paper we consider the Schr\"odinger operator $\mathcal L_V= -\Delta + V$ in $\mathbb R^d$ with a non negative potential $V$, and $V\not\equiv 0$. We define the logarithmic Schr\"odinger operator $\log \mathcal L_V$ proving its main…

We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem $L_\gamma u-\lambda u=\frac{u^{2^*(s)-1}}{|x|^s}$ on a smooth bounded domain $\Omega$ in $\mathbb{R}^n$…

The Special Lagrangian Potential Equation for a function $u$ on a domain $\Omega\subset {\bf R}^n$ is given by ${\rm tr}\{\arctan(D^2 \,u) \} = \theta$ for a contant $\theta \in (-n {\pi\over 2}, n {\pi\over 2})$. For $C^2$ solutions the…

We prove the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} L u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in }…

We prove existence and uniqueness of viscosity solutions to the degenerate parabolic problem $u_t = \Delta_\infty^h u$ where $\Delta_\infty^h$ is the $h$-homogeneous operator associated with the infinity-Laplacian, $\Delta_\infty^h u =…

In this paper we consider the following critical nonlocal problem $$ \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-\Delta)^s u =…

In the present paper, we study large values of Dirichlet $L$- functions inside the critical strip. For every $1/2<\sigma<1$, we show that for $q$ sufficiently large, there exists a non-principal character $\chi$ modulo $q$ and a constant…

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…