Splitting Type Variational Problems with Linear Growth Conditions
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Journal of Mathematical Sciences, Vol. 250, No. 2, October, 2020
SPLITTING TYPE VARIATIONAL PROBLEMS WITH LINEAR GROWTH CONDITIONS M. Bildhauer
∗
Universit¨ at des Saarlandes, Fachrichtung Mathematik Postfach 15 11 50, D–66041 Saarbr¨ ucken, Germany [email protected]
M. Fuchs Universit¨ at des Saarlandes, Fachrichtung Mathematik Postfach 15 11 50, D–66041 Saarbr¨ ucken, Germany [email protected]
UDC 517.958
Regularity properties of solutions to variational problems are established for a broad class of strictly convex splitting type energy densities of the principal form f : R2 → R, f (ξ1 , ξ2 ) = f1 (ξ1 ) + f2 (ξ2 ), with linear growth. We show that, regardless of the corresponding property of f2 , the assumption (t ∈ R) c1 (1 + |t|)−μ1 f1 (t) c2 ,
1 < μ1 < 2,
is sufficient to obtain higher integrability of ∂1 u for any finite exponent. We also include a series of variants of our main theorem. In the case f : Rn → R, similar results hold with the obvious changes in notation. Bibliography: 30 titles.
1
Introduction
In this paper, we discuss variational problems of linear growth with densities which do not belong to the class of μ-elliptic energies introduced first in [1]. Guided by linear growth examples of splitting type, which to our knowledge are not systematically studied up to now, we are led to quite general assumptions which still guarantee some interesting higher regularity properties of generalized solutions. Before going into details, we fix the framework of our consideration: in what follows, Ω denotes a bounded Lipschitz domain in Rn , n 2, u0 : Ω → R is a function such that (1.1) u0 ∈ W 1,2 (Ω) ∩ L∞ (Ω). Using a suitable approximation (cf., for example, [2] for more details), it is possible to assume that u0 ∈ W 1,1 (Ω) ∩ L∞ (Ω). ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 105, 2020, pp. 45-58. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2502-0232
232
We are interested in the variational problem J[u] := f (∇u) dx → min
in u0 + W01,1 (Ω)
(1.2)
Ω
for a strictly convex energy density f : Rn → [0, ∞) of class C 2 satisfying a1 |ξ| − a2 f (ξ) a3 |ξ| + a4 ,
ξ ∈ Rn ,
(1.3)
with suitable constants a1 , a3 > 0, a2 , a4 0. The condition (1.3) causes the well-known problems concerning the existence and regularity of solutions to the problem (1.2), which means that (1.2) should be replaced by a relaxed variant. For the general framework of this approach we refer, for example, to the monographs [3]–[7], where there are a lot of further references as well as a definition of the underlying spaces such as Lp (Ω), W 1,p (Ω), BV(Ω) and their local variants. According to [6, Theorem 5.47], the natural extension of (1.2) reads as ∇s w K[w] := f (∇a w) dx + f∞ d|∇s w| |∇s w| Ω
+
Ω
f∞ ((u0 − w)N )dH n−1 → min
in BV(Ω).
(1.4)
∂Ω
Here, ∇w = ∇a wL n + ∇s w is the Lebesgue decomposition of the vector measure ∇w with respect to the n-dimensional Lebesgue measure L n , f∞ is the recession function of f , i.
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