Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems

Teffera M Asfaw

doi:10.1155/2015/357934

Abstract

LetXbe a real locally uniformly convex reflexive separable Banach space with locally uniformly convex dual spaceX∗. LetT:X⊇D(T)→2X∗be maximal monotone andS:X⊇D(S)→X∗quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach spaceW⊂D(S), dense and continuously embedded inX. Assume, further, that there existsd≥0such that〈v∗+Sx,x〉≥-dx2for allx∈D(T)∩D(S)andv∗∈Tx. New surjectivity results are given for noncoercive, not everywhere defined, and possibly unbounded operators of the typeT+S. A partial positive answer for Nirenberg's problem on surjectivity of expansive mapping is provided. Leray-Schauder degree is applied employing the method of elliptic superregularization. A new characterization of linear maximal monotone operatorL:X⊇D(L)→X∗is given as a result of surjectivity ofL+S, whereSis of type(M)with respect toL. These results improve the corresponding theory for noncoercive and not everywhere defined operators of pseudomonotone type. In the last section, an example is provided addressing existence of weak solution inX=Lp(0,T;W01,p(Ω))of a nonlinear parabolic problem of the typeut-∑i=1n(∂/∂xi)ai(x,t,u,∇u)=f(x,t), (x,t)∈Q;u(x,t)=0, (x,t)∈∂Ω×(0,T);u(x,0)=0, x∈Ω, wherep>1,Ωis a nonempty, bounded, and open subset ofRN,ai:Ω×(0,T)×R×RN→R (i=1,2,…,n)satisfies certain growth conditions, andf∈Lp′(Q),Q=Ω×(0,T), andp′is the conjugate exponent ofp.

Highlights

PreliminariesIn what follows, X is a real reflexive separable locally uniformly convex Banach space with locally uniformly convex dual space X∗
A mapping T : X ⊃ D(T) → Y is “demicontinuous” if it is continuous from the strong topology of D(T) to the weak topology of Y
The following corollary gives a characterization of linear maximal monotone operator in separable reflexive Banach space

Summary

Introduction

PreliminariesIn what follows, X is a real reflexive separable locally uniformly convex Banach space with locally uniformly convex dual space X∗. Let T : X ⊇ D(T) → 2X∗ be maximal monotone and S : X ⊇ D(S) → X∗ quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach space W ⊂ D(S), dense and continuously embedded in X.

Results

Conclusion