Operators In Divergence Form Research Articles

Qualitative Analysis of General Elliptic Operator in Divergence Form

Qualitative Analysis of General Elliptic Operator in Divergence Form

On a nonlinear general eigenvalue problem in Musielak–Orlicz spaces

Abstract In this paper, we concern the existence result of the following general eigenvalue problem: { 𝒜 ⁢ ( u ) = λ ⁢ ℬ ⁢ ( u ) in ⁢ Ω , D α ⁢ ( u ) = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}\mathcal{A}(u)&\displaystyle={\lambda}% \mathcal{B}(u)&&\displaystyle\phantom{}\text{in }{\Omega},\\ \displaystyle D^{\alpha}(u)&\displaystyle=0&&\displaystyle\phantom{}\text{on }% {\partial\Omega},\end{aligned}\right. in an arbitrary Musielak–Orlicz spaces, where 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} are quasilinear operators in divergence form of order 2 ⁢ n {2n} and 2 ⁢ ( n - 1 ) {2(n-1)} , respectively. The main assumptions in this case are that 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} are potential operators with 𝒜 {\mathcal{A}} being elliptic and monotone. In this study, we intentionally avoid imposing constraints on the growth of a generalized N-function, including the Δ 2 {\Delta_{2}} -condition for both the generalized N-function and its conjugate. Consequently, this necessitates the formulation of the approximation theorem and the extensive utilization of modular convergence concepts.

Large deviation principle for pseudo-monotone evolutionary equation

. In this article, we explore Freidlin-Wentzell’s large deviation principle for a stochastic partial differential equation with the nonlinear diffusion-convection operator in divergence form satisfying p-type growth, involving coercivity assumptions, perturbed by small multiplicative Brownian noise. We use the weak convergence method to prove the Laplace principle, which is equivalent to the large deviation principle in our framework.

Counterexamples to maximal regularity for operators in divergence form

In this paper, we present counterexamples to maximal Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-regularity on H-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^{-1}$$\\end{document} under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-regularity on H-1(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^{-1}(\\mathbb {R}^d)$$\\end{document} or L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-regularity on L2(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\mathbb {R}^d)$$\\end{document}.

Harnack inequality and maximum principle for degenerate Kolmogorov operators in divergence form

We prove the parabolic strong maximum principle and the Harnack inequality for classical solutions to degenerate second order partial differential equations of the formLu:=∑i,j=1m0∂xi(aij∂xju)+∑j=1m0bj∂xju+cu+∑i,j=1Nbijxj∂xiu−∂tu=f, where 1≤m0≤N, A0=(aij)i,j=1,…,m0 is a bounded, symmetric and uniformly positive matrix and the matrix B:=(bij)i,j=1,…,N has real constant entries. Moreover, we assume low regularity (i.e. Hölder continuity) on the coefficients aij, bj and c, for i,j=1,…,m0. We point out the proofs of our main results only rely on the classical theory developed for harmonic functions.

Robust nonlocal trace spaces and Neumann problems

We prove trace and extension results for fractional Sobolev spaces of order s∈(0,1). These spaces are used in the study of nonlocal Dirichlet and Neumann problems on bounded domains. The results are robust in the sense that the continuity of the trace and extension operators is uniform as s approaches 1 and our trace spaces converge to H1/2(∂Ω). We apply these results in order to study the convergence of solutions of nonlocal Neumann problems as the integro-differential operators localize to a symmetric, second order operator in divergence form.

Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part

In the present paper we study perturbation theory for the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} Dirichlet problem on bounded chord arc domains for elliptic operators in divergence form with potentially unbounded antisymmetric part in BMO. Specifically, given elliptic operators L0=div(A0∇)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_0 = \ ext {div}(A_0\ abla )$$\\end{document} and L1=div(A1∇)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1 = \ ext {div}(A_1\ abla )$$\\end{document} such that the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} Dirichlet problem for L0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_0$$\\end{document} is solvable for some p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document}; we show that if A0-A1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_0 - A_1$$\\end{document} satisfies certain Carleson condition, then the Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document} Dirichlet problem for L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} is solvable for some q≥p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q \\ge p$$\\end{document}. Moreover if the Carleson norm is small then we may take q=p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=p$$\\end{document}. We use the approach first introduced in Fefferman–Kenig–Pipher ’91 on the unit ball, and build on Milakis–Pipher–Toro ’11 where the large norm case was shown for symmetric matrices on bounded chord arc domains. We then apply this to solve the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} Dirichlet problem on a bounded Lipschitz domain for an operator L=div(A∇)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L = \ ext {div}(A\ abla )$$\\end{document}, where A satisfies a Carleson condition similar to the one assumed in Kenig–Pipher ’01 and Dindoš–Petermichl–Pipher ’07 but with unbounded antisymmetric part.

Regularity theory and Green’s function for elliptic equations with lower order terms in unbounded domains

We consider elliptic operators in divergence form with lower order terms of the form Lu = -{{textrm{div}}}(A cdot nabla u + b u ) - c cdot nabla u - du, in an open set Omega subset mathbb {R}^n, n ge 3, with possibly infinite Lebesgue measure. We assume that the n times n matrix A is uniformly elliptic with real, merely bounded and possibly non-symmetric coefficients, and either b, c in L^{n,infty }_{text {loc}}({Omega }) and d in L_{text {loc}}^{frac{n}{2}, infty }(Omega ), or |b|^2, |c|^2, |d| in mathcal {K}_{text {loc}}(Omega ), where mathcal {K}_{text {loc}}(Omega ) stands for the local Stummel–Kato class. Let {mathcal {K}_{text {Dini}}}(Omega ) be a variant of mathcal {K}(Omega ) satisfying a Carleson-Dini-type condition. We develop a De Giorgi/Nash/Moser theory for solutions of Lu = f - {{textrm{div}}}g, where f and |g|^2 in {mathcal {K}_{text {Dini}}}(Omega ) if, for q in [n, infty ), any of the following assumptions holds: (i) |b|^2, |d| in {mathcal {K}_{text {Dini}}}(Omega ) and either c in L^{n,q}_{text {loc}}(Omega ) or |c|^2 in mathcal {K}_{text {loc}}(Omega ); (ii) {{textrm{div}}}b +d le 0 and either b+c in L^{n,q}_{text {loc}}(Omega ) or |b+c|^2 in mathcal {K}_{text {loc}}(Omega ); (iii) -{{textrm{div}}}c + d le 0 and |b+c|^2 in {mathcal {K}_{text {Dini}}}(Omega ). We also prove a Wiener-type criterion for boundary regularity. Assuming global conditions on the coefficients, we show that the variational Dirichlet problem is well-posed and, assuming -{{textrm{div}}}c +d le 0, we construct the Green’s function associated with L satisfying quantitative estimates. Under the additional hypothesis |b+c|^2 in mathcal {K}'(Omega ), we show that it satisfies global pointwise bounds and also construct the Green’s function associated with the formal adjoint operator of L. An important feature of our results is that all the estimates are scale invariant and independent of Omega , while we do not assume smallness of the norms of the coefficients or coercivity of the associated bilinear form.

G-convergence of elliptic operators in non divergence form in R^n

The aim of this note is to prove a characterization of the G-limit of a sequence of elliptic operators in non-divergence form. As we consider any dimension, for this class of operators, it is not enough to deal with measurable and bounded coefficients so we need extra regularity assumptions on them.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/71/abstr.html

Inequalities for eigenvalues of operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

In this paper, we compute universal inequalities of eigenvalues of a large class of second-order elliptic differential operators in divergence form, that includes, e.g., the Laplace-Beltrami and Cheng-Yau operators, on a bounded domain in a complete Riemannian manifolds isometrically immersed in Euclidean space. A key step in order to obtain the sequence of our estimates is to get the right Yang-type first inequality. We also prove some inequalities for manifolds supporting some special functions and tensors.

Elliptic operators with discontinuous coefficients in meshfree GFDM

AbstractIn phase change simulations, material properties such as density, viscosity, or thermal conductivity may exhibit jump discontinuities, possibly of several orders of magnitude. These jump discontinuities represent interfaces between the phases, and they emerge naturally during the simulation; thus, their exact location is generally unknown a priori. Our goal is to simulate phase change processes with a meshfree generalized finite difference method in a monolithic model without distinguishing between the different phases. There, the material properties mentioned above appear as coefficients inside elliptic operators in divergence form and the jumps must be treated adequately by the numerical method. We present a numerical method for discretizing elliptic operators with discontinuous coefficients without the need for a domain decomposition or tracking of interfaces. Our method facilitates the construction of diagonally dominant diffusion operators that lead to M‐matrices for the discrete Poisson's equation, and thus, satisfy the discrete maximum principle. We demonstrate the applicability of the new method for the case of smooth diffusivity and discontinuous diffusivity. We show that the method is first‐order accurate for discontinuous diffusion problems and provides second‐order and fourth‐order convergence for continuous diffusion coefficients.

On approximation of the Dirichlet problem for divergence form operators by Robin problems

We show that, under natural assumptions, solutions of Dirichlet problems for uniformly elliptic divergence form operators can be approximated pointwise by solutions of some versions of Robin problems. The proof is based on the stochastic representation of solutions and properties of reflected diffusions corresponding to divergence form operators.

Trilinear embedding for divergence-form operators with complex coefficients

We prove a dimension-free Lp(Ω)×Lq(Ω)×Lr(Ω)→L1(Ω×(0,∞)) embedding for triples of elliptic operators in divergence form with complex coefficients and subject to mixed boundary conditions on Ω, and for triples of exponents p,q,r∈(1,∞) mutually related by the identity 1/p+1/q+1/r=1. Here Ω is allowed to be an arbitrary open subset of Rd. Our assumptions involving the exponents and coefficient matrices are expressed in terms of a condition known as p-ellipticity. The proof utilizes the method of Bellman functions and heat flows. As a corollary, we give applications to (i) paraproducts and (ii) square functions associated with the corresponding operator semigroups, moreover, we prove (iii) inequalities of Kato–Ponce type for elliptic operators with complex coefficients. All the above results are the first of their kind for elliptic divergence-form operators with complex coefficients on arbitrary open sets. Furthermore, the approach to (ii),(iii) through trilinear embeddings seems to be new.

Gradient estimates for the insulated conductivity problem with inclusions of the general m‐convex shapes

AbstractIn this paper, the insulated conductivity model with two touching or close‐to‐touching inclusions is considered in with . We establish the pointwise upper bounds on the gradient of the solution for the generalized m‐convex inclusions under these two cases with , which show that the singular behavior of the gradient in the thin gap between two inclusions is described by the first non‐zero eigenvalue of an elliptic operator of divergence form on . Finally, the sharpness of the estimates is also proved for two touching axisymmetric insulators, especially including curvilinear cubes.

Eigenvalue estimates for a class of elliptic differential operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

Eigenvalue estimates for a class of elliptic differential operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

Dynamic boundary conditions for divergence form operators with Hölder coefficients

Dynamic boundary conditions for divergence form operators with Hölder coefficients

Non-linear operators of divergence form on the Sierpinski gasket

We consider non-linear operators L of divergence type on the Sierpinski gasket. We prove the Harnack inequality for non-negative L-harmonic functions on the Sierpinski gasket. The proof is based on Moser’s method, and we construct a new measure λ by combining a Hausdorff measure and an energy measure, which plays an essential role at key steps. The Harnack inequality is achieved by involving the Caccioppoli type inequality and the weighted Sobolev inequality which give the local boundedness and the Weak Harnack inequality of solutions with respect to the measure λ.

Structure of Green’s function of elliptic equations and helical vortex patches for 3D incompressible Euler equations

We develop a new structure of the Green’s function of a second-order elliptic operator in divergence form in a 2D bounded domain. Based on this structure and the theory of rearrangement of functions, we construct concentrated traveling-rotating helical vortex patches to 3D incompressible Euler equations in an infinite pipe. By solving an equation for vorticity $$\begin{aligned} w=\frac{1}{\varepsilon ^2}f_\varepsilon \left( \mathcal {G}_{K_H}w-\frac{\alpha }{2}|x|^2|\ln \varepsilon |\right) \ \ \text {in}\ \Omega \end{aligned}$$ for small $$ \varepsilon >0 $$ and considering a certain maximization problem for the vorticity, where $$ \mathcal {G}_{K_H} $$ is the inverse of an elliptic operator $$ \mathcal {L}_{K_H} $$ in divergence form, we get the existence of a family of concentrated helical vortex patches, which tend asymptotically to a singular helical vortex filament evolved by the binormal curvature flow. We also get nonlinear orbital stability of the maximizers in the variational problem under $$ L^p $$ perturbation when $$ p\ge 2. $$

Bilinear embedding in Orlicz spaces for divergence-form operators with complex coefficients

We prove a bi-sublinear embedding for semigroups generated by non-smooth complex-coefficient elliptic operators in divergence form and for certain mutually dual pairs of Orlicz-space norms. This generalizes a result by Carbonaro and Dragičević from power functions to more general Young functions that still behave like powers. To achieve this, we generalize a Bellman function constructed by Nazarov and Treil.

Wolff potentials and measure data vectorial problems with Orlicz growth

We study solutions to measure data elliptic systems with Uhlenbeck-type structure that involve operator of divergence form, depending continuously on the spacial variable, and exposing doubling Orlicz growth with respect to the second variable. Pointwise estimates for the solutions that we provide are expressed in terms of a nonlinear potential of generalized Wolff type. Not only we retrieve the recent sharp results proven for p-Laplace systems, but additionally our study covers the natural scope of operators with similar structure and natural class of Orlicz growth.