Linear and nonlinear abstract equations with parameters

This paper focuses on nonlocal boundary value problems for linear and nonlinear abstract elliptic equations in Banach spaces. Here equations and boundary conditions contain certain parameters. The uniform separability of the linear problem and the existence and uniqueness of maximal regular solution of nonlinear problem are obtained in L_p spaces. For linear case the discreteness of spectrum of corresponding parameter dependent differential operator is obtained. The behavior of solution when the parameter approaches zero and its smoothness with respect to the parameter is established. Moreover, we show the estimate for analytic semigroups in terms of interpolation spaces. This fact can be used to obtain maximal regularity properties for abstract boundary value problems.     

Coercive boundary value problems for regular degenerate differential-operator equations

This study focuses on nonlocal boundary value problems (BVP) for degenerate elliptic differential-operator equations (DOE), that are defined in Banach-valued function spaces, where boundary conditions contain a degenerate function and a principal part of the equation possess varying coefficients. Several conditions obtained, that guarantee the maximal L_p regularity and Fredholmness. These results are also applied to nonlocal BVP for regular degenerate partial differential equations on cylindrical domain to obtain the algebraic conditions that ensure the same properties.     

Free Boundary Value Problems for Abstract Elliptic Equations and Applications

The free boundary value problems for elliptic differential-operator equations are studied. Several conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract L _p -spaces are given. In application, the nonlocal free boundary problems for finite or infinite systems of elliptic and anisotropic type equations are studied.     

A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations

We consider nonlinear elliptic equations driven by the p-Laplacian differential operator. Using degree theoretic arguments based on the degree map for operators of type (S)₊ , we prove theorems on the existence of multiple nontrivial solutions of constant sign.     

Weak solutions for nonlocal boundary value problems with low regularity data

This paper concerns the existence and uniqueness of weak solutions for elliptic and parabolic equations under nonlocal boundary conditions, based on maximal regularity. It also gives the positivity of solutions which can be used in monotone iteration methods. As an application, the results are used to discuss some specific nonlocal problems.     

Abstract Elliptic Equations with Integral Boundary Conditons

This paper focuses on nonlocal integral boundary value problems for elliptic differential-operator equations. Here given conditions guarantee that maximal regularity and Fredholmness in $L_{p}$ spaces. These results are applied to the Cauchy problem for abstract parabolic equations, its infinite systems and boundary value problems for anisotropic partial differential equations in mixed $L_{\mathbf{p}}$ norm.     

On the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and Dirichlet boundary conditions

We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains with Dirichlet boundary conditions. We impose mild conditions on the domains and lower order (nonlinear) coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by Kato class functions. Our approach uses an implicit probabilistic representation, Schauder's fixed point theorem, and new a priori estimates for solutions of the corresponding linear elliptic equations. In the course of deriving these a priori estimates we show that the Green functions for operators of the form on D are comparable when one modifies the drift term b on a compact subset of D. This generalizes a previous result of Ancona [2], obtained under an condition on b, to a Kato condition on . Received: 21 April 1998 / in final form 26 March 1999     

Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces

We consider coerciveness and Fredholmness of nonlocal boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. In some special cases, the main coefficients of the boundary conditions may be bounded operators and not only complex numbers. Then, we prove an isomorphism, in particular, maximal L _p -regularity, of the problem with a linear parameter in the equation. In both cases, the boundary conditions may also contain unbounded operators in perturbation terms. Finally, application to regular nonlocal boundary value problems for elliptic equations of the second order in non-smooth domains are presented. Equations and boundary conditions may contain differential-integral parts. The spaces of solvability are Sobolev type spaces W _p,q ^2,2. The first author is a member of G.N.A.M.P.A. and the paper fits the 60% research program of G.N.A.M.P.A.-I.N.D.A.M.; The third author was supported by the Israel Ministry of Absorption.     

Smoothness of generalized solutions for higher-order elliptic equations with nonlocal boundary conditions

The smoothness of generalized solutions for higher-order elliptic equations with nonlocal boundary conditions is studied in plane domains. Necessary and sufficient conditions upon the right-hand side of the problem and nonlocal operators under which the generalized solutions possess an appropriate smoothness are established.     

Irregular boundary value problems for second order elliptic differential-operator equations in UMD Banach spaces

We prove coerciveness with a defect and Fredholmness of nonlocal irregular boundary value problems for second order elliptic differential-operator equations in UMD Banach spaces. Then, we prove coerciveness with a defect in both the space variable and the spectral parameter of the problem with a linear parameter in the equation. The results do not imply maximal L _p -regularity in contrast to previously considered regular case. In fact, a counterexample shows that there is no maximal L _p -regularity in the irregular case. When studying Fredholmness, the boundary conditions may also contain unbounded operators in perturbation terms. Finally, application to nonlocal irregular boundary value problems for elliptic equations of the second order in cylindrical domains are presented. Equations and boundary conditions may contain differential-integral parts. The spaces of solvability are Sobolev type spaces ${W_{p,q}^{2,2}}$ .     

Maximal B-regular boundary value problems with parameters

This study focuses on non-local boundary value problems (BVP) for elliptic differential-operator equations (DOE) defined in Banach-valued Besov (B) spaces. Here equations and boundary conditions contain certain parameters. This study found some conditions that guarantee the maximal regularity and fredholmness in Banach-valued B-spaces uniformly with respect to these parameters. These results are applied to non-local boundary value problems for a regular elliptic partial differential equation with parameters on a cylindrical domain to obtain algebraic conditions that guarantee the same properties.     

Nonlocal limits in the study of linear elliptic systems arising in periodic homogenization

In the present paper, we obtain the two-scale limit system of a sequence of linear elliptic periodic problems with varying coefficients. We show that this system has not the same structure than the classical one, obtained when the coefficients are fixed. This is due to the apparition of nonlocal effects. Our results give an example showing that the homogenization of elliptic problems with varying coefficients, depending on one parameter, gives in general a nonlocal limit problem.     

Separable anisotropic elliptic operators and applications

This paper focuses on boundary value problems for anisotropic differential-operator equations of high order with variable coefficients in the half plane. Several conditions are obtained which guarantee the maximal regularity of anisotropic elliptic and parabolic problems in Banach-valued L _p -spaces. Especially, it is shown that this differential operator is R-positive and is a generator of an analytic semigroup. These results are also applied to infinite systems of anisotropic type partial differential equations in the half plane.     

Second Order Elliptic Differential-Operator Equations with Unbounded Operator Boundary Conditions in <Emphasis Type="Italic">UMD</Emphasis> Banach Spaces

In a UMD Banach space E, we consider a boundary value problem for a second order elliptic differential-operator equation with a spectral parameter when one of the boundary conditions, in the principal part, contains a linear unbounded operator in E. A theorem on an isomorphism is proved and an appropriate estimate of the solution with respect to the space variable and the spectral parameter is obtained. In this way, Fredholm property of the problem is shown. Moreover, discreteness of the spectrum and completeness of a system of root functions corresponding to the homogeneous problem are established. Finally, applications of obtained abstract results to nonlocal boundary value problems for elliptic differential equations with a parameter in non-smooth domains are given.     

Maximum principle and existence of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms

We study L^p-viscosity solutions of fully nonlinear, second-order, uniformly elliptic partial differential equations (PDE) with measurable terms and quadratic nonlinearity. We present a sufficient condition under which the maximum principle holds for L^p-viscosity solution. We also prove stability and existence results for the equations under consideration.     

Quasilinear parabolic and elliptic equations with nonlinear boundary conditions

This paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both Dirichlet and nonlinear Neumann boundary conditions. The equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient. The consideration of the class of equations is motivated by some heat-transfer problems where the heat capacity and thermal conductivity are both temperature dependent. The aim of the paper is to show the existence and uniqueness of a global time-dependent solution of the parabolic problem, existence of maximal and minimal steady-state solutions of the elliptic problem, including conditions for the uniqueness of a solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions. Applications are given to some heat-transfer problems and an extended logistic reaction–diffusion equation.     

Projektive Newton-Verfahren bei elliptischen Randwertaufgaben

Summary In this paper we investigate projective Newton methods for nonlinear elliptic boundary value problems. These methods yield approximations by solving the linear equations of the Newton method by a projection method, e.g. the Ritz method. Using subspaces of finite elements or polynominals we obtain error estimates and optimal convergence theorems (inH ₁ andL ₂).

     

Boundary behaviour for solutions of elliptic singular equations with a gradient term

We investigate the homogeneous Dirichlet boundary value problem for a class of second-order nonlinear elliptic partial differential equations with a gradient term and singular data. Under general conditions on the data, we study the behaviour of the solution near the boundary of the domain. Under suitable additional conditions we also investigate the second-order term in the asymptotic expansion of the solution in terms of the distance from the boundary.     

Nondegeneracy of elliptic problems with periodic nonlinearities in the plane

A class of asymptotically quadratic functionals on Hilbert spaces, called degenerate, is considered and explored. Our results are applied to obtain a extension, in the planar case, of a result published by Solimini in ‘On the solvability of some elliptic partial differential equations with the linear part at resonance’, J. Math. Anal. Appl., 117 (1986), 138-152. Similar extensions had been previously studied in the literature only for domains with particular geometries.     

Existence of solutions to quasi-linear elliptic problems with generalized Robin boundary conditions

We characterize the L^q-solvability of a class of quasi-linear elliptic equations involving the p-Laplace operator with generalized nonlinear Robin type boundary conditions on bad domains. Some uniqueness results are also given.