On interior and boundary regularity of weak solutions to a certain quasilinear elliptic system

Jakobowsky, Norbert

doi:10.1007/bf02570815

Cited by 5 publications

(6 citation statements)

References 12 publications

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“…In this generality that result, apparently, seems to be unknown in the existing literature. Under more restrictive assumptions on H, and using a different method, a similar theorem has been obtained by Jakobowsky [19].…”

Section: 1) Is Hölder Continuous On Compact Subsets Of Ibmentioning

confidence: 70%

A new proof of regularity of weak solutions of the H -surface equation

Strzelecki

2003

Calculus of Variations and Partial Differential Equations

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We give a new proof of a theorem of Bethuel, asserting that arbitrary weak solutions u ∈ W 1,2 (IB, IR 3 ) of the H-surface system ∆u = 2H(u)ux 1 ∧ ux 2 are locally Hölder continuous provided that H is a bounded Lipschitz function. Contrary to Bethuel's, our proof completely omits Lorentz spaces. Estimates below natural exponents of integrability are used instead. (The same method yields a new proof of Hélein's theorem on regularity of harmonic maps from surfaces into arbitrary compact Riemannian manifolds.) We also prove that weak solutions with continuous trace are continuous up to the boundary, and give an extension of these results to the equation of hypersurfaces of prescribed mean curvature in IR n+1 , this time assuming in addition that |∇H(y)| decays at infinity like |y| −1 .

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Section: 1) Is Hölder Continuous On Compact Subsets Of Ibmentioning

confidence: 70%

A new proof of regularity of weak solutions of the H -surface equation

Strzelecki

2003

Calculus of Variations and Partial Differential Equations

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“…In the next section we will use several times the following boundary regularity result for solutions to the Dirichlet problem of (2.2). The result comes from Jacobowsky's paper [Jac,Theorem 4.1] (see also the paper by Brezis and Coron [BrCo]).…”

Section: A Boundary Regularity Resultsmentioning

confidence: 99%

Surfaces of constant curvature inR3with isolated singularities

Gálvez¹,

Hauswirth²,

Mira³

2013

Advances in Mathematics

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We prove that finite area isolated singularities of surfaces with constant positive curvature K > 0 in R 3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally convex curves in S 2 with admissible cusp singularities, characterizing when the singularity is actually embedded. In the global setting, we describe the space of peaked spheres in R 3 , i.e. compact convex surfaces of constant curvature K > 0 with a finite number of singularities, and give applications to harmonic maps and constant mean curvature surfaces.

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“…Inferring ideas of Heinz [7] and the author [10], [11] used for regularity results, we deduce uniform estimates for the logarithmically weighted Dirichlet integrals. (2.7), (2.8)).…”

Section: Strong Convergence Inmentioning

confidence: 93%

“…Similarly to [10], [11], we insert (l -<t> 2e , s (\x" -<f w ,fe,)l) 3 ) + * 2 «..(l*,. is non-negative and can be dropped.…”

Section: N'mentioning

confidence: 99%

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A perturbation result concerning a second solution to the Dirichlet problem for the equation of prescribed mean curvature.

Jakobowsky¹

1994

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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A perturbation result concerning a second solution to the Dirichlet problem for the equation of prescribed mean curvature By Norbert Jakobowsky at Aachen 0. Introduction Let B = {(u, v) e IR 2 : u 2 + v 2 < 1}. We look for mappings : B -> R 3 solving the Dirichlet problem (DP) for the equation of prescribed mean curvaturewhere H is a given real bounded function in IR 3 , the exterior product in R 3 , and z 0 denotes the boundary data. Solutions to (DP) can be obtained äs critical points of the functional3 )nL°° and divß(x) = 3H(x). As a suitable choice for Q we set (0.3) Q(x) = ß H (x) = (7 H (s, x 2 , xjds, ] H(x i9 s, x 3 )ds, 7 Ä(^, x 2 , s)ds] , which reduces to ß(x) = /ix in case of H = const, and define E H = E Q . Generalizing results of Heinz [5], and Werner [27], for H = const, Hildebrandt [8], [9] found that under certain conditions on z 0 and H the functional E H admits some relative minimizer: Theorem 1. (i) Suppose z 0 e W^2(B 9 R 3 )nL°°, He (0.4) H^olioo;B

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On interior and boundary regularity of weak solutions to a certain quasilinear elliptic system

Cited by 5 publications

References 12 publications

A new proof of regularity of weak solutions of the H -surface equation

A new proof of regularity of weak solutions of the H -surface equation

Surfaces of constant curvature inR3with isolated singularities

A perturbation result concerning a second solution to the Dirichlet problem for the equation of prescribed mean curvature.

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