Related papers: Green representations and global H\"older continui…

This article is about the $\mathbb{Z}^d$-periodic Green function $G_n(x,y)$ of the multiscale elliptic operator $Lu=-{\rm div}\left( A(n\cdot) \cdot \nabla u \right)$, where $A(x)$ is a $\mathbb{Z}^d$-periodic, coercive, and H\"older…

In this paper we deal with the following boundary value problem \begin{equation*} \begin{cases} -\Delta_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in $\Omega$,} \newline u\geq 0 & \text{in $\Omega$,} \newline u=0 & \text{on $\partial…

We show that bounded solutions of the quasilinear elliptic equation $\Delta_{p(x)} u=g+div(\textbf{F})$ are locally H\"{o}lder continuous provided that the functions $g$ and $\textbf{F}$ are in suitable Lebesgue spaces.

Let $\Gamma$ be geometric tree graph with $m$ edges and consider the second order Sturm-Liouville operator $\L[u]=(-pu')'+qu$ acting on functions that are continuous on all of $\Gamma$, and twice continuously differentiable in the interior…

We establish existence and pointwise estimates of fundamental solutions and Green's matrices for divergence form, second order strongly elliptic systems in a domain $\Omega \subseteq \mathbb{R}^n$, $n \geq 3$, under the assumption that…

In this paper we prove a H\"older partial regularity result for weak solutions $u:\Omega\to \mathbb{R}^N$, $N\geq 2$, to non-autonomous elliptic systems with general growth of the type: \begin{equation*} -\rm{div}\, a(x, u, Du)= b(x, u, Du)…

We study existence and stability of solutions of (E 1) --$\Delta$u + $\mu$ |x| 2 u + g(u) = $\nu$ in $\Omega$, u = 0 on $\partial$$\Omega$, where $\Omega$ is a bounded, smooth domain of R N , N $\ge$ 2, containing the origin, $\mu$ $\ge$ --…

In this paper, we characterize all the distributions $F \in \mathcal{D}'(U)$ such that there exists a continuous weak solution $v \in C(U,\mathbb{C}^{n})$ (with $U \subset \Omega$) to the divergence-type equation…

In two dimensions every weak solution to a nonlinear elliptic system $\rm{div} a(x,u,Du)=0$ has H\"older continuous first derivatives provided that standard continuity, ellipticity and growth assumptions hold with a growth exponent $p \geq…

We prove that $L^2$ weak solutions to hypoelliptic equations with bounded measurable coefficients are H\"older continuous. The proof relies on classical techniques developed by De Giorgi and Moser together with the averaging lemma and…

In this paper, we investigate the regularity of weak solutions $u\colon\Omega\to\mathbb{R}$ to elliptic equations of the type \begin{equation*} \mathrm{div}\, \nabla \mathcal{F}(x,Du) = f\qquad\text{in $\Omega$}, \end{equation*} whose…

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation \[ \mathcal{L}u = \sigma u^{q} + \mu \quad \text{in} \;\; \Omega, \] in the sublinear case $0<q<1$, with…

We obtained the $C^\alpha$ continuity for weak solutions of a class of ultraparabolic equations with measurable coefficients of the form $${\partial_t u} = \partial_x(a(x,y,t)\partial_x u) +b_0(x,y,t)\partial_x u+b(x,y,t)\partial_y u,$$…

If h is a ring-valued function on a simplicial complex G we can define two matrices L and g, where the matrix entries are the h energy of homoclinic intersections. We know that the sum over all h values on G is equal to the sum of the Green…

For a given second order elliptic operation $\mathcal{L}$ in a domain $\Omega\subset{\mathbb{R}}^\mathbf{N}$, $\mathbf{N}\ $, and a compact set $\mathbf{K}\subset\Omega$, order $\mathbf{N}$-$2$-Ahlfors-David regular, we define the space…

Let $L$ be a second order elliptic operator $L$ with smooth coefficients defined on a domain $\Omega $ in $\mathbb{R}^d $, $d\geq3$, such that $L1\leq 0$. We study existence and properties of continuous solutions to the following problem…

Here we obtain bounds on the spectrum of that operator whose inverse, when it exists, gives the Green's function. We consider the wide of physical problems that can be cast in a form where a constitutive equation ${\bf J}({\bf x})={\bf…

We prove the local H\"older regularity of weak solutions to the mixed local nonlocal parabolic equation of the form \begin{equation*} u_t-\Delta u+\text{P.V.}\int_{\mathbb{R}^{n}} {\frac{u(x,t)-u(y,t)}{{\left|x-y\right|}^{n+2s}}}dy=0,…

We study the local H\"older regularity of strong solutions $u$ of second-order uniformly elliptic equations having a gradient term with superquadratic growth $\gamma > 2$, and right-hand side in a Lebesgue space $L^q$. When $q >…

We prove a multiplicity result for non-constant weak solutions $u \in H^1(\Omega)$ for the quasilinear elliptic equation \[ \begin{cases} \displaystyle-\text{div}(A(x,u)\nabla u) + \frac{1}{2} D_sA(x,u)\nabla u \cdot \nabla u = g(x,u) -…