Ein optimaler Regularit�tssatz f�r schwache L�sungen gewisser elliptischer Systeme

Wiegner, Michael

doi:10.1007/bf01214271

Cited by 44 publications

(16 citation statements)

References 2 publications

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“…Note, that a similar discussion concerning the growth factor holds for the elliptic analogue; references for C α -estimates and -regularity under rather optimal smallness condition are [13,17]. Note further, that, without C α -estimates, in general it is false that a sequence of (possibly approximate) solutions to (1.2), which is bounded in L ∞ (L 2 )∩L 2 (H 1 ), has a subsequence which converges strongly in L 2 (H 1 ).…”

Section: Introductionmentioning

confidence: 92%

Smooth Solutions of systems of quasilinear parabolic equations

Bensoussan

Frehse²

2002

ESAIM: COCV

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Abstract. We consider in this article diagonal parabolic systems arising in the context of stochastic differential games. We address the issue of finding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic differential games. Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear Hamiltonian seems to be the adequate one to achieve the regularity property. A key step in the theory is to prove the existence of Hölder solution.

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Section: Introductionmentioning

confidence: 92%

Smooth Solutions of systems of quasilinear parabolic equations

Bensoussan

Frehse²

2002

ESAIM: COCV

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“…First results can be found in the book of Ladyzhenskaya and Uralt'seva [24]. Then it was shown by Hildebrandt, Widman and Wiegner [19,39,38] that n ≥ 3 and I 1 imply Hölder continuity of a given weak solution (and in the scalar case actually any bounded solution is regular). A generalization under condition I 4 was then given in [20], and a proof based on a geometric approach in [6].…”

Section: Introductionmentioning

confidence: 98%

Regular and irregular solutions for a class of elliptic systems in the critical dimension

Beck

Frehse

2012

Nonlinear Differ. Equ. Appl.

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Abstract.We study regularity properties of weak solutions in the Sobolev space W 1,n 0 to inhomogeneous elliptic systems under a natural growth condition and on bounded Lipschitz domains in R n , i. e. we investigate weak solutions in the limiting situation of the Sobolev embedding. Several counterexamples of irregular solutions are constructed in cases, where additional structure conditions might have led to regularity. Among others we present both bounded irregular and unbounded weak solutions to elliptic systems obeying a one-sided condition, and we further construct unbounded extremals of two-dimensional variational problems. These counterexamples do not exclude the existence of a regular solution. In fact, we establish the existence of regular solutions-under standard assumptions on the principal part and the aforementioned one-sided condition on the inhomogeneity. This extends previous works for n = 2 to more general cases, including arbitrary dimensions. Moreover, this result is achieved by a simplified proof invoking modern techniques. Mathematics Subject Classification (2000). Primary 35J47, 35B65; Secondary 35A01.

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“…Also in that section, we finish to prove the main result, Theorem 1. The last section, §7, is devoted to the analysis of problem (3), (4). Here Theorem 4 is proved, which is the main partial regularity result for the solution of that problem.…”

Section: Introductionmentioning

confidence: 99%

“…The obstacle problem for quadratic functionals of the form (1), (5) and under the Dirichlet boundary condition has been studied by Hildebrandt, Widman, Fuchs, Duzaar, Wiegner and other authors (see [4,5,6,7,8,9,10,11,12,13,14] and the references therein). Various obstacles of the type u(Ω) ⊂ K have been treated, where K ⊂ R N .…”

Section: Introductionmentioning

confidence: 99%

“…In particular, it was shown that dim H Σ ≤ n − 3, and Σ may only consist of isolated points if n = 3 and K is a compact subset of R N with C 3 -smooth boundary ∂K (see [7]). Various restrictions on K of a geometric nature have been stated under which the solution of the variational problem with obstacle is smooth on s Ω (see [13,14,4,11]). In the author's paper [15], partial regularity up to the boundary was proved for functions that provide the minimal value for the functional (1), (5) in the case of noncompact K with ∂K ∈ C 2 .…”

Section: Introductionmentioning

confidence: 99%

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A problem with an obstacle that goes out to the boundary of the domain for a class of quadratic functionals on $\mathbb{R}^{N}$

Arkhipova

2011

St. Petersburg Math. J.

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Abstract. A variational problem with obstacle is studied for a quadratic functional defined on vector-valued functions u : Ω → R N , N > 1. It is assumed that the nondiagonal matrix that determines the quadratic form of the integrand depends on the solution and is "split". The role of the obstacle is played by a closed (possibly, noncompact) set K in R N or a smooth hypersurface S. It is assumed that u(x) ∈ K or u(x) ∈ S a.e. on Ω. This is a generalization of a scalar problem with an obstacle that goes out to the boundary of the domain. It is proved that the solutions of the variational problems in question are partially smooth in s Ω and that the singular set Σ of the solution satisfies H n−2 (Σ) = 0.

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Ein optimaler Regularit�tssatz f�r schwache L�sungen gewisser elliptischer Systeme

Cited by 44 publications

References 2 publications

Smooth Solutions of systems of quasilinear parabolic equations

Smooth Solutions of systems of quasilinear parabolic equations

Regular and irregular solutions for a class of elliptic systems in the critical dimension

A problem with an obstacle that goes out to the boundary of the domain for a class of quadratic functionals on $\mathbb{R}^{N}$

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