Elliptic equations with critical nonlinear exponent

In [1, 2, 3, 4, 6], the authors posed and discussed some boundary‐value problems of second order elliptic equations with parabolic degeneracy. In [5], the authors posed and discussed some boundary‐value problems of second order mixed equations with degenerate curve on the sides of an angle, but the coefficients of the equations possess strong restrictions. The present paper deals with the mixed problem for elliptic equations with degenerate curve on the sides of an angle, where the coefficients satisfy general conditions. We first give the formulation of the problem and estimates of solutions of the problem for the equations, and then prove the existence of solutions for the above problem by the Leray–Schauder theorem. In this paper, we use the complex analytic method, namely we first introduce the new notation (2.1) below and reduce the degenerate elliptic equations of second order to the corresponding problem for degenerate elliptic complex equations of first order, afterwards the above problem of second order degenerate elliptic equations can be solved. The results in this paper will be used in a subsequent paper to handle the Tricomi problem of second order equations of mixed type with degenerate curve on the sides of an angle. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper we discuss some problems about critical point theory. In the first part of the paper we study existence and multiplicity results of semilinear second order elliptic equation: $$ \begin{cases} -\Delta u=f(x,u) &\text{for } x\in \Omega, \\ u=0 &\text{for } x\in \partial \Omega, \end{cases} $$ In [Z. Li. Liu and S. J. Li, Contractibility of level sets of functionals associated with some elliptic boundary value problems and applications , NoDEA 10 (2003), 133–170], the authors study the contractibility of level sets of functionals associated with some elliptic boundary value problems. In this paper by using Morse theory and minimax method we give a more precise description of topological construction of level set of critical value of energy functional for mountain pass type critical point. It is well known that nondegenerate critical point is isolated, so if a critical point is not isolated, it must be a degenerate critical point. In the second part we will give an example that all the critical points of functional of a class of oscillating equation with Neumann boundary condition are isolated and the equation has only constant solutions. Moreover, critical groups of each critical point of the functional are trivial. The elliptic sine-Gordon equation originates from the static case of the hyperbolic sine-Gordon equation modelling the Josephson junction in superconductivity, which is of contemporary interest to physicists. The problem is similar to the elliptic sine-Gordon equation so we believe that it derives from profound physical backdrop.

In this paper, we investigate a two‐grid weak Galerkin method for semilinear elliptic differential equations. The method mainly contains two steps. First, we solve the semilinear elliptic equation on the coarse mesh with mesh size , then, we use the coarse mesh solution as an initial guess to linearize the semilinear equation on the fine mesh, that is, on the fine mesh (with mesh size ), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution has sufficient regularity and , the two‐grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.

We generalize the notion of renormalized solution to semilinear elliptic and\nparabolic equations involving operator associated with general (possibly\nnonlocal) regular Dirichlet form and smooth measure on the right-hand side. We\nshow that under mild integrability assumption on the data a quasi-continuous\nfunction $u$ is a renormalized solution to an elliptic (or parabolic) equation\nin the sense of our definition iff $u$ is its probabilistic solution, i.e. $u$\ncan be represented by a suitable nonlinear Feynman-Kac formula. This implies in\nparticular that for a broad class of local and nonlocal semilinear equations\nthere exists a unique renormalized solution.\n

The aim of this chapter is to discuss some full (that is everywhere) regularity results for scalar-valued weak solutions to second order elliptic equations in divergence forms. Weak solutions only belong to some Sobolev space (consequently, neither are they necessarily continuous nor do derivatives a priori exist in the classical sense), and hence, their regularity needs to be investigated. We here prove local Holder regularity of weak solution and present two different (and classical) strategies of proof dating back to the late 1950s. First, we explain De Giorgi’s level set technique, in a unified approach that applies both to weak solutions of elliptic equations and to minimizers of variational integrals, via the study of Q-minimizers of suitable functionals. We then address, for the specific case of linear elliptic equations, an alternative proof of the everywhere regularity result of weak solutions via Moser’s iteration method.

The aim of this thesis is to prove the existence of a weak solution for semilinear fractional elliptic equations given by \begin{eqnarray*}\left\{ \begin{array}{ll} (-\Delta)^s u=|u|^{p-2} u+ f(x),& \quad x\in \Omega,\\ u=0, & \quad x \in \R^n \backslash \Omega, %0, & \quad x \in \partial \Oemga, \end{array} \right. \end{eqnarray*} where $(-\Delta)^s$ denotes the fractional Laplace operator with $s \in (0,1],$ $n > 2s,$ $\Omega$ is an open bounded domain in $\R^n$ with $C^2$-boundary and $f \in L^2(\Omega).$ We are interested in extending this result for $s \in (0, 1]$ to $p$ greater than the critical Sobolev exponent where the compact embedding fails to hold. We shall make use of a new variational principle presented in \cite{Mo1} that allows one to deal with problems well beyond the compact structure.

We demonstrate a new nonuniform mesh finite difference method to obtain accurate solutions for the elliptic partial differential equations in two dimensions with nonlinear first-order partial derivative terms. The method will be based on a geometric grid network area and included among the most stable differencing scheme in which the nine-point spatial finite differences are implemented, thus arriving at a compact formulation. In general, a third order of accuracy has been achieved and a fourth-order truncation error in the solution values will follow as a particular case. The efficiency of using geometric mesh ratio parameter has been shown with the help of illustrations. The convergence of the scheme has been established using the matrix analysis, and irreducibility is proved with the help of strongly connected characteristics of the iteration matrix. The difference scheme has been applied to test convection diffusion equation, steady state Burger’s equation, ocean model and a semi-linear elliptic equation. The computational results confirm the theoretical order and accuracy of the method.

In this paper we study the uniform approximation of continuous functions on compact subsets of R n by solutions of second order elliptic equations. Also we discuss the problem of existence of a continuous best approximation and the connection of the approximation problem to some isoperimetric inequalities.KeywordsMaximum PrincipleElliptic EquationDirichlet ProblemUniform ApproximationIsoperimetric InequalityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Necessary and sufficient conditions are established for the existence of limits in the sense, 1$ SRC=http://ej.iop.org/images/0025-5734/48/2/A19/toc_tex_sm_2693_img2.gif/>, on the boundary of a domain, of solutions of second order elliptic equations in domains with Lyapunov boundaries.Bibliography: 8 titles.

In this paper we show how a second order scalar uniformly elliptic equation in divergence form with measurable coefficients and Dirichlet boundary conditions can be transformed into a first order elliptic system with half-Dirichlet boundary condition. This first order system involves Hodge-Dirac operators and can be seen as a natural generalization of the Beltrami equation in the plane and we develop a theory for this equation, extending results from the plane to higher dimension. The reduction to a first order system applies both to linear as well as quasilinear second order equations and we believe this to be of independent interest. Using the first order system, we give a new representation formula of the solution of the Dirichlet problem both on simply and finitely connected domains. This representation formula involves only singular integral operators of convolution type and Neumann series there of, for which classical Calderón-Zygmund theory is applicable. Moreover, no use is made of any fundamental solution or Green's function beside fundamental solutions of constant coefficient operators. Remarkably, this representation formula applies also for solutions of the fully non-linear first order system. We hope that the representation formula could be used for numerically solving the equations. Using these tools we give a new short proof of Meyers' higher integrability theorem. Furthermore, we show that the solutions of the first order system are Hölder continuous with the same Hölder coefficient as the solutions of the second order equations. Finally, factorization identities and representation formulas for the higher dimensional Beurling-Ahlfors operator are proven using Clifford algebras, and certain integral estimates for the Cauchy transform is extended to higher dimensions.

This paper contains new uniqueness results of the boundary blow-up viscosity solutions of second order elliptic equations, generalizing a well known result of Marcus-Veron for the Laplace operator.

In this paper we study the topological properties of the level sets,$ \S_{t}(u)=\left\{x:~u(x)= t \right\}$, of solutions $u$of second order elliptic equations with vanishing zeroth order terms. Weshow that the total Betti number of level sets $\S_{t}$ is a uniformlybounded function of the parameter $t$. The uniform bound can be estimatedin terms of the analytic coefficients as well as the generalized degreesof the corresponding solutions. Such an estimate is also valid for thenodal sets of solutions of the same type equations with zerothorder terms. In general, it is possible to derive from our analysis anestimate for the total Betti numbers of level sets, for large measure setof $t's$, when coefficients are sufficiently smooth, and therefore a $L^{p}$bound on Betti numbers as a function of $t$. These estimatesare obtained by a quantitative Stability Lemma and a quantitative MorseLemma.

Let Ω be an open set in high dimension Euclidean spaces R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , and let L denote the second order elliptic operator in the general form, which will characterize many of the original phenomenon of nature. As a very useful space, Orlicz space L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Φ</sup> includes many classical spaces such as L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> space, where 1< p < ∞ and Φ is a convex increasing function. The purpose of this paper is to use the stopping time technique and establish the regularity estimates in Orlicz spaces for the second order derivatives of the solutions to the general second order elliptic equations and their Dirichlet problems. We will obtain that, if u is the solution of the Dirichlet problem to the equation Lu = f for any given data f, then the weak solution u as well as its gradient ∇u and second order derivatives ∇ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> u can be controlled by the data f even in the Orlicz spaces.

This paper deals with elliptic equations in the plane with degen- eracies. The equations are generated by a complex vector field that is elliptic everywhere except along a simple closed curve. Kernels for these equations are constructed. Properties of solutions, in a neighborhood of the degen- eracy curve, are obtained through integral and series representations. An application to a second order elliptic equation with a punctual singularity is given.

A method for the numerical solution of the time dependent two-dimensional magneto-plasma dynamic coupled problem in the low magnetic Reynolds number case, is presented. The method is based on a time dependent two-dimensional description of the fluid dynamics iteratively coupled to a steady state two-dimensional description of the electrodynamics. This is allowed as the characteristic times of variation of the electrodynamic quantities are much lower than those of the fluid dynamic quantities. The fluid is assumed to be compressible and viscous, and governed by the Navier-Stokes equations. A semi-implicit finite difference pressure scheme has been utilised for the solution of the fluid dynamic problem. The electrodynamics is described by a second order elliptic partial differential equation obtained from the Maxwell equations and generalised Ohm's law. The elliptic equation is solved by means of an exponential fitting finite difference method. In order to test the algorithm described above, the transient of a plasma flow in a channel, caused by the variation of the externally applied magnetic field, has been studied.