Hölder continuity conditions for the solvability of Dirichlet problems involving functionals with free discontinuities (original) (raw)

Regularity of minimizers for a class of anisotropic free discontinuity problems

2001

This paper contains existence and regularity results for solutions u : Ω → R n N of a class of free discontinuity problems i.e.: the energy to minimize consists of both a bulk and a surface part. The main feature of the class of problems considered here is that the energy density of the bulk part is supposed to be fully anisotropic with p-growth in the scalar case, n = 1. Similar results for the vectorial case n > 1 are obtained for radial energy densities, being anisotropic again with p-growth.

Approximation of free � discontinuity problems by elliptic functionals via � convergence

2003

defined on the space SBV (Ω) of special functions of bounded variation are studied, where ∇u denotes the approximate gradient of u, and Su is the set of the discontinuity points of u. In a two-dimensional setting, Su represents the contours of the object in a picture and u is a smoothing of an imput image. Energies of the same form arise in fracture mechanics for brittle solids, where Su is interpreted as the crack surface and u as the displacement outside the fractured region ([4]). Problems involving functionals of this form are usually called free-discontinuity problems, after a terminology introduced by De Giorgi (see [11], [5], [7]). The Ambrosio and Tortorelli approach [6] provides a variational approximation of the Mumford and Shah functional (1) via elliptic functionals to obtain approximate smooth solutions and overcome the numerical problems due to surface detection. The unknown surface Su is substituted by an additional function variable v which approaches the characteris...

Continuity of the Free Boundary in Elliptic Problems with Neuman Boundary Condition

2015

We show the continuity of the free boundary in a class of two dimensional free boundary problems with Neuman boundary condition, which includes the aluminium electrolysis problem and the heterogeneous dam problem with leaky boundary condition. 1. Statement of the problem and preliminary results Let Ω be the open bounded domain of R defined by Ω = {(x1, x2) ∈ R : x1 ∈ (a0, b0), d0 < x2 < γ(x1)} where a0, b0, d0 are real numbers and γ is a real-valued Lipschitz continuous function on (a0, b0). Let a(x) = (aij(x)) be a two-by-two matrix and h a function defined in Ω with aij ∈ L∞(Ω), |a(x)| ≤ Λ, for a.e. x ∈ Ω, (1.1) a(x)ξ · ξ ≥ λ|ξ| ∀ξ ∈ R, for a.e. x ∈ Ω, (1.2) h ≤ h(x) ≤ h̄ for a.e. x ∈ Ω (1.3) hx2 ∈ L p loc(Ω) (1.4) hx2(x) ≥ 0 for a.e. x ∈ Ω. (1.5) where λ, Λ, h̄, h and p are positive constants such that h̄ ≥ h and p > 2. Let Γ = {(x1, γ(x1)) : x1 ∈ (a0, b0)} and let β(x, u) be a function defined on Γ× R satisfying β(x, .) is Lipschitz continuous for a.e. x ∈ Γ (1.6) β(x, ...

Non-locality, non-linearity, and existence of solutions to the Dirichlet problem for least gradient functions in metric measure spaces

2022

We study the Dirichlet problem for least gradient functions for domains in metric spaces equipped with a doubling measure and supporting a (1,1)-Poincar\'e inequality when the boundary of the domain satisfies a positive mean curvature condition. In this setting, it was shown by Mal\'y, Lahti, Shanmugalingam, and Speight that solutions exist for continuous boundary data. We extend these results, showing existence of solutions for boundary data that is approximable from above and below by continuous functions. We also show that for each finL1(partialOmega),f\in L^1(\partial\Omega),finL1(partialOmega), there is a least gradient function in Omega\OmegaOmega whose trace agrees with fff at points of continuity of fff, and so we obtain existence of solutions for boundary data which is continuous almost everywhere. This is in contrast to a result of Spradlin and Tamasan, who constructed an L1L^1L1-function on the unit circle which has no least gradient solution in the unit disk in mathbbR2.\mathbb{R}^2.mathbbR2. Modifying the example of Spradlin...

The Dirichlet problem by variational methods

Bulletin of the London Mathematical Society, 2008

Let Ω ⊂ R N be an bounded open set and ϕ ∈ C(∂Ω). Assume that ϕ has an extension Φ ∈ C(Ω) such that ∆Φ ∈ H −1 (Ω). Then by the Riesz representation theorem there exists a unique u ∈ H 1 0 (Ω) such that − ∆u = ∆Φ in D(Ω) ′. We show that u + Φ coincides with the Perron solution of the Dirichlet problem ∆h = 0, h| ∂Ω = ϕ. This extends recent results by Hildebrandt [Math. Nachr. 278 (2005), 141-144] and Simader [Math. Nachr. 279 (2006), 415-430], and also gives a possible answer to Hadamard's objection against Dirichlet's principle.

The Dirichlet problem in domains with lower dimensional boundaries

Revista Matemática Iberoamericana, 2020

The present paper pioneers the study of the Dirichlet problem with L q boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in Ω := R n \ R d with d < n − 1. Following the first results in [DFM1], [DFM2], the authors introduce an appropriate degenerate elliptic operator and show that the Dirichlet problem is solvable for all q > 1 provided that the coefficients satisfy the small Carleson norm condition. Even in the context of the classical case d = n − 1, (the analogues of) our results are new. The conditions on the coefficients are more relaxed than the previously known ones (most notably, we do not impose any restrictions whatsoever on the first n − 1 rows of the matrix of coefficients) and the results are more general. We establish local rather than global estimates between the square function and the non-tangential maximal function and, perhaps even more importantly, we establish new Moser-type estimates at the boundary and improve the interior ones.