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The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values. (English) Zbl 1120.46016

The authors consider the Dirichlet energy integral

Ω |u(x)| p(x) dx

with variable p(x), where Ω is a bounded domain in n , which was studied by various authors. The existence and regularity of minimizers for this integral were studied by E. Acerbi and G. Minigone [Arch. Ration. Mech. Anal. 156, 121–140 (2001; Zbl 0984.49020)] for functions uW 1,1 (Ω) with boundary values in the classical sense, the minimizer being Hölder continuous in this case. The authors develop another approach to the study of this integral, based on the paper by N. Shanmugalingam [Ill. J. Math. 45, No. 3, 1021–1050 (2001; Zbl 0989.31003)].

They admit functions with boundary values in the Sobolev sense and minimize over functions in the variable exponent Sobolev space. To this end, they define the Sobolev space W 0 1,p(·) with zero boundary values in terms of the Sobolev variable capacity introduced earlier by the authors. As a preliminary step, they also study a variable Poincaré p(·)-inequality and introduce a condition on p(x) under which such an inequality is valid.

On this basis, they prove their main results in Theorems 5.2 and 5.3. In the former, they show that if 1<essinfp(x)esssupp(x)< and p(x) “is not too discontinuous”, then there exists a function uW 1,p(·) (Ω) which minimizes the Dirichlet p(·)-integral with u-wW 0 1,p(·) (Ω). The minimizer is unique in a certain sense. In the latter, they give a criterion for a function u to be a minimizer. The results obtained are parallel to those known in the case of constant p.

46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
49J40Variational methods including variational inequalities
31C45Nonlinear potential theory, etc.
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