A weighted estimate of the gradient for nonnegative generalized solutions of quasilinear doubly degenerate parabolic equations

For nonnegative generalized solutions of doubly degenerate parabolic equations, a weighted estimate of the form, is establised.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 3–23, 1990.     

Branches of positive solutions of quasilinear elliptic equations on non-smooth domains

We prove existence of an unbounded global branch (i.e. connected set) of weak solutions of a second order quasilinear equation depending on a real parameter λ$\lambda$ on an arbitrary (possibly non-smooth) bounded domain in R^N${\mathbb{R}}^N$, with a Leray–Lions operator as the leading part. Here, we can allow lower order nonlinearities which depend on first derivatives, satisfying appropriate growth conditions including the critical case. Furthermore, we give sufficient conditions for the existence of a branch consisting entirely of nonnegative solutions for positive λ$\lambda$. Our approach also yields a new existence result in the case of critical growth in derivatives of lower order.     

Isolated singularities for weighted quasilinear elliptic equations

We classify all the possible asymptotic behavior at the origin for positive solutions of quasilinear elliptic equations of the form div(|∇u|^p−2∇u)=b(x)h(u) in Ω?{0}, where 1<p?N and Ω is an open subset of RN with 0∈Ω. Our main result provides a sharp extension of a well-known theorem of Friedman and Véron for h(u)=uq and b(x)≡1, and a recent result of the authors for p=2 and b(x)≡1. We assume that the function h is regularly varying at ∞ with index q (that is, limt→∞h(λt)/h(t)=λq for every λ>0) and the weight function b(x) behaves near the origin as a function b₀(|x|) varying regularly at zero with index θ greater than −p. This condition includes b(x)=θ|x| and some of its perturbations, for instance, b(x)=θ|x|m(−log|x|) for any m∈R. Our approach makes use of the theory of regular variation and a new perturbation method for constructing sub- and super-solutions.     

Boundary element methods for Maxwell's equations on non-smooth domains

Summary. Variational boundary integral equations for Maxwell's equations on Lipschitz surfaces in are derived and their well-posedness in the appropriate trace spaces is established. An equivalent, stable mixed reformulation of the system of integral equations is obtained which admits discretization by Galerkin boundary elements based on standard spaces. On polyhedral surfaces, quasioptimal asymptotic convergence of these Galerkin boundary element methods is proved. A sharp regularity result for the surface multipliers on polyhedral boundaries with plane faces is established. Received January 5, 2001 / Revised version received August 6, 2001 / Published online December 18, 2001 Correspondence to: C. Schwab     

An estimate for the solutions of stokes equations in exterior domains

A boundary value problem for the Stokes equations is examined in an exterior domain ⁿ with a uniform Dirichlet condition on the boundary and a homogeneous condition at infinity. It is shown that estimating the norm L _p() of the second derivatives of the velocity vector field by the same norm of the exterior forces vector field is correct for p < n/2, but not for p n/2. This estimate is valid also for p n/2 if the boundary conditions are modified at infinity. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akademii Nauk SSSR, Vol. 180, pp. 105–120, 1990.     

A sharp gradient estimate for the weighted p-Laplacian

Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, d?? = e ^h (x) dV (x) the weighted measure and ??_??,p the weighted p-Laplacian. In this paper we consider the non-linear elliptic equation $$ Delta _{mu ,p} u = - lambda _{mu ,p} |u|^{p - 2} u $$ for p ?? (1, 2). We derive a sharp gradient estimate for positive smooth solutions of this equation. As applications, we get a Harnack inequality and a Liouville type theorem..     

Classical solutions for the pressure-gradient equations in non-smooth and non-convex domains

We establish the existence of the classical solution for the pressure-gradient equation in a non-smooth and non-convex domain. The equation is elliptic inside the domain, becomes degenerate on the boundary, and is singular at the origin when the origin lies on the boundary. We show the solution is smooth inside the domain and continuous up to the boundary.     

A global a posteriori error estimate for quasilinear elliptic problems

A global a posteriori error estimate, valid even if uniqueness fails, is obtained for a class of quasilinear elliptic partial differential equations. It is applied to the analysis of finite element methods for such problems, and to a primitive Fourier series method for the stationary Navier-Stokes problem in three dimensions at arbitrary Reynolds number.This research was supported by the National Science Foundation, Grant No. NSF-GP-37069.     

Global weighted estimates for quasilinear elliptic equations with non-standard growth

In this paper we obtain a new global gradient estimates in weighted Lorentz spaces for weak solutions of p(x)$p(x)$-Laplacian type equation with small BMO coefficients in a δ-Reifenberg flat domain. The modified Vitali covering lemma, the maximal function technique and the appropriate localization method are the main analytical tools. Our results improve the known results for such equations.     

A priori estimate for a quasilinear problem depending on the gradient

The main purpose of this paper is to establish a priori estimate for positive solutions of some superlinear, quasilinear elliptic equations where the nonlinearity depends on x, u, and ∇u. Our argument does not need a non-existence result for the limit problem obtained by the usual blow-up procedure. This work is related to a previous one by Ruiz (2004) [9].     

On quasilinear parabolic equations involving weighted p-Laplacian operators

In this paper we consider the initial boundary value problem for a class of quasilinear parabolic equations involving weighted p-Laplacian operators in an arbitrary domain, in which the conditions imposed on the non-linearity provide the global existence, but not uniqueness of solutions. The long-time behavior of the solutions to that problem is considered via the concept of global attractor for multi-valued semiflows. The obtained results recover and extend some known results related to the p-Laplacian equations.     

Mixed boundary value problems for nonlinear elliptic equations in multidimensional non-smooth domains

The nonlinear elliptic equation is investigated. It is supposed that u fulfils a mixed boundary value condition and that Ω ? IRⁿ (n ≥ 3) has a piecewise smooth boundary. W^s,2 — regularity (s < 3/2) of u and L^p — properties of the first and the second derivatives of u are proven.     

On global bifurcation for quasilinear elliptic systems on bounded domains

General second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated into nonlinear mappings between Sobolev spaces. It is shown that the linearized mapping is a Fredholm operator of index zero. This and the abstract global bifurcation theorem of [Jacobo Pejsachowicz, Patrick J. Rabier, Degree theory for C¹ Fredholm mappings of index 0, J. Anal. Math. 76 (1998) 289-319] allow us to carry out bifurcation analysis directly on these elliptic systems. At the abstract level, we establish a unilateral global bifurcation result that is needed when studying positive solutions. Finally, we supply two examples of cross-diffusion population model and chemotaxis model to demonstrate how the theory can be applied.     

On quasilinear parabolic evolution equations in weighted L p -spaces

In this paper we develop a geometric theory for quasilinear parabolic problems in weighted L _p -spaces. We prove existence and uniqueness of solutions as well as the continuous dependence on the initial data. Moreover, we make use of a regularization effect for quasilinear parabolic equations to study the ω-limit sets and the long-time behaviour of the solutions. These techniques are applied to a free boundary value problem. The results in this paper are mainly based on maximal regularity tools in (weighted) L _p -spaces.     

Global attractors for quasilinear parabolic equations on unbounded thin domains

Monatshefte für Mathematik - In this paper we are concerned with the asymptotic behavior of quasilinear parabolic equations posed in a family of unbounded domains that degenerates onto a lower...