how to solve super position theorem
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Topological Methods in Nonlinear Analysis
Journal of the Juliusz Schauder Center
Volume 28, 2006, 87–103
SOME GENERAL CONCEPTS
OF SUB- AND SUPERSOLUTIONS
FOR NONLINEAR ELLIPTIC PROBLEMS
Vy Khoi Le — Klaus Schmitt
Abstract. We propose general and unified concepts of sub- supersolutions
for boundary value problems that encompass several types of boundary con-
ditions for nonlinear elliptic equations and variational inequalities. Various,
by now classical, sub- and supersolution existence and comparison results
are covered by the general theory presented here.
1. Introduction — Problem settings
We are interested here in sub-supersolution results for boundary value prob-
lems with second order principal operators and general boundary conditions.
The problems may or may not contain obstacles or constraints. Based on the
weak (variational) formulation of the problem, we deduce that the boundary
conditions (or at least parts of them) may usually be encoded into the set of test
(admissible) functions.
The goal of this paper is to show that in several cases (covering those that
have been studied in the literature), by formulating the problem as a variational
inequality, even if it is a smooth equation, we may give simple, unified, and
general definitions of sub- and supersolutions. These concepts of sub- and su-
persolutions extend the classical definitions for equations subject to Dirichlet,
2000 Mathematics Subject Classification. 35B45, 35J65, 35J60.
Key words and phrases. Sub- and supersolutions, general boundary conditions, variational
inequalities.
c 2006 Juliusz Schauder Center for Nonlinear Studies
Explanation:
V. K. Le — K. Schmitt
Neumann, Robin, or No-Flux (periodic boundary conditions for the one space
dimensional problem) boundary conditions (see e.g. [16]) and are motived by the
recent definitions of sub-supersolutions for variational inequalities in [11], [13],
[14]. Also, we can demonstrate the existence of solutions and extremal solutions
between sub- and supersolutions and other properties of the solution sets when
sub- and supersolutions exist.
Let Ω ⊂ R
N be a bounded domain with Lipschitz boundary and W1,p(Ω) be
the usual first-order Sobolev space with the norm
(1.1) kuk = kukW1,p(Ω) = (kuk
p
Lp(Ω) + k|∇u|kp
Lp(Ω))
1/p, u ∈ W1,p(Ω).
Assume that K is a closed, convex subset of W1,p(Ω). We consider the following
variational inequality on K:
(1.2)
Z
Ω
A(x, ∇u) · (∇v − ∇u) dx +
Z
Ω
f(x, u)(v − u) dx
+
Z
∂Ω
g(x, u)(v − u) dS ≥ 0, for all v ∈ K,
u ∈ K.
We remark that in order to simplify the notation we use u and v instead of u|∂Ω
and v|∂Ω for the trace of u and v on ∂Ω in the surface integral in (1.2). This
simplification will also be used in the sequel in other instances when it is clear
from the context. In the variational inequality (1.2), A is an elliptic operator, f
is the lower order term, and g is a boundary term.
Problems such as (1.2), in the case of (smooth) equations, i.e. K is a sub-
space of W1,p(Ω), have been studied by sub-supersolution methods in, e.g. [7],
[5], [16] and some of the references therein, subject to different boundary condi-
tions (usually homogeneous ones). In previous papers, sub- and supersolutions
are defined using inequality conditions on the boundary. Therefore, different
boundary conditions require different definitions of sub- and supersolutions. As
a consequence, separate arguments and calculations are needed to study the ex-
istence and properties of solutions between sub- and supersolutions. In what
follows, we show that common, unified definitions of sub- and supersolutions
may be given for various types of boundary conditions (including unilateral con-
straints). Thus a common, comprehensive general existence theorem is possible
for many different types of boundary value problems. The sub-supersolution
approach for variational inequalities with homogeneous Dirichlet boundary con-
ditions was studied in a systematic way in [13]. Our results here are motivated
by and also generalize those in the papers [11], [13], [14].
We begin with the assumptions on the principal operator. Assume that
A: Ω × R
N → R