Existence and boundedness of minimizers of a class of integral functionals

A. Mercaldo

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 1, page 125-139
  • ISSN: 0392-4041

Abstract

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In this paper we consider a class of integral functionals whose integrand satisfies growth conditions of the type f ( x , η , ξ ) ≥ a ( x ) | ξ | p ( 1 + | η | ) α - b 1 ( x ) | η | β 1 - g 1 ( x ) , f ( x , η , 0 ) ≤ b 2 ( x ) | η | β 2 + g 2 ( x ) , where 0 ≤ α < p , 1 ≤ β 1 < p , 0 ≤ β 2 < p , α + β i ≤ p , a ⁢ x , b i ⁢ x , g i ⁢ x ( i = 1 , 2 ) are nonnegative functions satisfying suitable summability assumptions. We prove the existence and boundedness of minimizers of such a functional in the class of functions belonging to the weighted Sobolev space W 1 , p ⁢ a , which assume a boundary datum u 0 ∈ W 1 , p ⁢ a ∩ L ∞ ⁢ Ω .

How to cite

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Mercaldo, A.. "Existence and boundedness of minimizers of a class of integral functionals." Bollettino dell'Unione Matematica Italiana 6-B.1 (2003): 125-139. <http://eudml.org/doc/194751>.

@article{Mercaldo2003,
abstract = {In this paper we consider a class of integral functionals whose integrand satisfies growth conditions of the type \begin\{gather*\} f(x, \eta , \xi ) \ge a(x) \frac\{|\xi |^\{p\}\}\{(1 + |\eta |)^\{\alpha \}\} - b\_\{1\}(x)|\eta |^\{\beta \_\{1\}\}-g\_\{1\}(x),\\ f(x, \eta , 0)\le b\_\{2\}(x)|\eta |^\{\beta \_\{2\}\}+ g\_\{2\}(x), \end\{gather*\} where$0\leq \alpha <p$, $1\leq \beta_\{1\}< p$, $0\leq \beta_\{2\}< p$, $\alpha+\beta_\{i\}\leq p$, $a(x)$, $b_\{i\}(x)$, $g_\{i\}(x)$ ($i= 1$, $2$) are nonnegative functions satisfying suitable summability assumptions. We prove the existence and boundedness of minimizers of such a functional in the class of functions belonging to the weighted Sobolev space $W^\{1,p\}(a)$ , which assume a boundary datum $u_\{0\}\in W^\{1,p\}(a)\cap L^\{\infty\}(\Omega)$.},
author = {Mercaldo, A.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {125-139},
publisher = {Unione Matematica Italiana},
title = {Existence and boundedness of minimizers of a class of integral functionals},
url = {http://eudml.org/doc/194751},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Mercaldo, A.
TI - Existence and boundedness of minimizers of a class of integral functionals
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/2//
PB - Unione Matematica Italiana
VL - 6-B
IS - 1
SP - 125
EP - 139
AB - In this paper we consider a class of integral functionals whose integrand satisfies growth conditions of the type \begin{gather*} f(x, \eta , \xi ) \ge a(x) \frac{|\xi |^{p}}{(1 + |\eta |)^{\alpha }} - b_{1}(x)|\eta |^{\beta _{1}}-g_{1}(x),\\ f(x, \eta , 0)\le b_{2}(x)|\eta |^{\beta _{2}}+ g_{2}(x), \end{gather*} where$0\leq \alpha <p$, $1\leq \beta_{1}< p$, $0\leq \beta_{2}< p$, $\alpha+\beta_{i}\leq p$, $a(x)$, $b_{i}(x)$, $g_{i}(x)$ ($i= 1$, $2$) are nonnegative functions satisfying suitable summability assumptions. We prove the existence and boundedness of minimizers of such a functional in the class of functions belonging to the weighted Sobolev space $W^{1,p}(a)$ , which assume a boundary datum $u_{0}\in W^{1,p}(a)\cap L^{\infty}(\Omega)$.
LA - eng
UR - http://eudml.org/doc/194751
ER -

References

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