This paper deals with the $L^p$ 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 $L^p$ theory, developed in [1, 24] and [2], cannot be extended under any restriction on the essential range of the coefficients. Our constructions...

We establish the existence of a T-p(x)-solution for the p(x)-elliptic problem $-div\left(a\right(x,u,\nabla u\left)\right)+g(x,u)=f-divF$ in Ω, where Ω is a bounded open domain of ${ℝ}^N$, N ≥ 2 and $a:\Omega \times ℝ\times {ℝ}^N\to {ℝ}^N$ is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but with only a weak monotonicity condition. The right hand side f lies in L¹(Ω) and F belongs to ${\prod }_{i=1}^N{L}^{p^{\text{'}}\left(·\right)}\left(\Omega \right)$.

The strong closure of feasible states of families of operators is studied. The results are obtained for self-adjoint operators in reflexive Banach spaces and for more concrete case - families of elliptic systems encountered in the optimal layout of $r$ materials. The results show when it is possible to parametrize the strong closure by the same type of operators. The results for systems of elliptic operators for the case when number of unknown functions $m$ is less than the dimension $n$ of...

ContentsIntroduction.............................................................................................................................31. Definitions, notations and some auxiliary lemmas...................................................42. The definition of the spaces $H_{p,q;Y}(\Omega ,ℬ)$..........................................................73. Some properties of the spaces $H_{p,q;Y}(\Omega ,ℬ)$...................................................104. Some examples of the spaces $H_{p,q;Y}(\Omega ,ℬ)$....................................................155....

Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ $-{\Delta }_{p}u+{\left|u\right|}^{p-2}u=f_{1\lambda ₁}{\left(x\right)\left|u\right|}^{q-2}u+2\alpha /(\alpha +\beta )g_{\mu }{\left|u\right|}^{\alpha -2}u{\left|v\right|}^{\beta }$, x ∈ Ω, ⎨ $-{\Delta }_{p}v+{\left|v\right|}^{p-2}v=f_{2\lambda ₂}{\left(x\right)\left|v\right|}^{q-2}v+2\beta /(\alpha +\beta )g_{\mu }{\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -2}v$, x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and $\Omega \subset {ℝ}^N$ is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and $f_{i{\lambda }_i}\left(x\right)={\lambda }_i{f}_{i+}\left(x\right)+f_{i-}\left(x\right)$ (i = 1,2) are sign-changing functions, where $f_{i\pm }\left(x\right)=max\pm f_i\left(x\right),0$, $g_{\mu }\left(x\right)=a\left(x\right)+\mu b\left(x\right)$, and ${\Delta }_{p}u={div\left(\right|\nabla u|}^{p-2}\nabla u)$ denotes the p-Laplace operator. We use variational methods.

The study of the equation (L₂L₁)*h = 0 or of the equivalent system L*₂h₂ = -h₁, L*₁h₁ = 0, where $L_j(j=1,2)$ is a second order elliptic differential operator, leads us to the following general framework: Starting from a biharmonic space, for example the space of solutions (u₁,u₂) of the system L₁u₁ = -u₂, L₂u₂ = 0, $L_j(j=1,2)$ being elliptic or parabolic, and by means of its Green pairs, we construct the associated adjoint biharmonic space which is in duality with the initial one.

We study elliptic equations with the general nonstandard growth conditions involving Lebesgue measurable functions on $\Omega$. We prove the global $C^{1,\alpha }$ regularity of bounded weak solutions of these equations with the Dirichlet boundary condition. Our results generalize the $C^{1,\alpha }$ regularity results for the elliptic equations in divergence form not only in the variable exponent case but also in the constant exponent case.

By a sub-super solution argument, we study the existence of positive solutions for the system ⎧$-{\Delta }_{p}u=a₁\left(x\right)F₁(x,u,v)$ in Ω, ⎪$-{\Delta }_{q}v=a₂\left(x\right)F₂(x,u,v)$ in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in ${ℝ}^N$ with smooth boundary or $\Omega ={ℝ}^N$. A nonexistence result is obtained for radially symmetric solutions.

We study a general class of nonlinear elliptic problems associated with the differential inclusion $\beta \left(u\right)-div\left(a\right(x,Du)+F(u\left)\right)\ni f$ in Ω where $f\in L^{\infty }\left(\Omega \right)$. The vector field a(·,·) is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in function spaces we prove existence of renormalized solutions for general $L^{\infty }$-data.

We study the existence of positive solutions to ⎧ ${div\left(\right|\nabla u|}^{p-2}\nabla u)+q\left(x\right)u^{-\gamma }=0$ on Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω is ${ℝ}^N$ or an unbounded domain, q(x) is locally Hölder continuous on Ω and p > 1, γ > -(p-1).

We prove the local Hölder continuity of bounded generalized solutions of the Dirichlet problem associated to the equation ${\sum }_{i=1}^m\frac{\partial }{\partial x_i}{a}_i(x,u,\nabla u)-c_0{\left|u\right|}^{p-2}u=f(x,u,\nabla u),$ assuming that the principal part of the equation satisfies the following degenerate ellipticity condition $\lambda \left(\right|u\left|\right){\sum }_{i=1}^m{a}_i(x,u,\eta ){\eta }_i\ge \nu \left(x\right){\left|\eta \right|}^p,$ and the lower-order term $f$ has a natural growth with respect to $\nabla u$.

We discuss the existence of positive radial solutions of the semilinear elliptic equation ⎧-Δu = K(|x|)f(u), x ∈ Ω ⎨αu + β ∂u/∂n = 0, x ∈ ∂Ω, ⎩$li{m}_{\left|x\right|\to \infty }u\left(x\right)=0$, where $\Omega =x\in {ℝ}^N:\left|x\right|>r₀$, N ≥ 3, K: [r₀,∞) → ℝ⁺ is continuous and $0<{\int }_{r₀}^{\infty }rK\left(r\right)dr<\infty$, f ∈ C(ℝ⁺,ℝ⁺), f(0) = 0. Under the conditions related to the asymptotic behaviour of f(u)/u at 0 and infinity, the existence of positive radial solutions is obtained. Our conditions are more precise and weaker than the superlinear or sublinear growth conditions. Our discussion is based on the...

Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to $Lu=div\overrightarrow{f}$ in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the $L^q(\Omega ,d\mu )$ norm of |∇u| is dominated by the $L^p(\Omega ,dv)$ norms of $div\overrightarrow{f}$ and $|\overrightarrow{f}|$. If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.

We consider a random, uniformly elliptic coefficient field $a$ on the lattice ${\mathbb{Z}}^d$. The distribution $\langle ·\rangle$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ and $L^1$, respectively, in probability, i.e., they obtained bounds on $\langle |{\nabla }_{x}G(t,x,y){{|}^2\rangle }^{1/2}$ and $\langle \left|{\nabla }_x{\nabla }_{y}G(t,x,y)\right|\rangle$. In particular, the elliptic Green’s function $G(x,y)$ satisfies optimal annealed bounds. In their recent work,...

Let M be a $C^{\infty }$-smooth n-dimensional manifold and ν₁, ..., νₙ be $C^{\infty }$-smooth vector fields on M which span the tangent space $T_{x}M$ at each point x ∈ M. The vector fields ν₁, ..., νₙ may have nonzero commutators. We construct a calculus of pseudodifferential operators (ψDOs) which act on sections of vector bundles over M and have symbols belonging to anisotropic analogues of the Hörmander classes $S_{\varrho ,\delta }^r$, and apply it to semi-elliptic operators generated by ν₁,...,νₙ. The results obtained include the...

This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by $$\begin{array}{cccc}\hfill t2& -{\mathrm{div}\left(b\right(\left|u\right|\left)\right|\nabla u|}^{p-2}{\nabla u)+d(\left|u\right|\left)\right|\nabla u|}^p=f-{\mathrm{div}\left(c\left(x\right)\right|u|}^{\alpha })\hfill & \hfill & \text{in}\Omega ,\hfill \\ & u=0\hfill & \hfill & \text{on}\partial \Omega ,\hfill \end{array}t$$ where $\Omega$ is a bounded open set of ${ℝ}^N$ ($N\ge 2$) with $1<p<N$ and $f\in L^1\left(\Omega \right),$ under some growth conditions on the function $b(·)$ and $d(·),$ where $c(·)$ is assumed to be in $L^{\frac{N}{(p-1)}}\left(\Omega \right).$ We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.

For $0<p-1<q$ and either $ϵ=1$ or $ϵ=-1$, we prove the existence of solutions of $-{\Delta }_{p}u=ϵu^q$ in a cone $C_S$, with vertex 0 and opening $S$, vanishing on $\partial C_S$, of the form $u\left(x\right)={\left|x\right|}^{-\beta }\omega (x/\left|x\right|)$. The problem reduces to a quasilinear elliptic equation on $S$ and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.

A "matching" method for planar harmonic functions is exhibited, using an elliptic adjoint equation $L^{\star }u=0$.