A simple partial regularity proof for minimizers of variational integrals - Schmidt, Thomas (Nodea-Nonlinear Differential Equations and Applications, 2009)
We consider multi-dimensional variational integrals F[u] := ∫Omega; f(·, u, Du) dx for u : ℝn⊃ Ω→ ℝN, where the integrand f is a strictly convex function of its last argument. We give an elementary proof for the partial C1α -regularity of minimizers of F. Our approach is based on the method of A-harmonic approximation, avoids the use of Gehring's lemma, and establishes partial regularity with the optimal Hölder exponent α in a single step. © 2009 Birkhaueser.
UnivIS-Import:2015-03-09:Pub.2009.nat.dma.lma2.asimpl
Keywords: Minimizer; Partial regularity; Variational integral

Asymptotically regular problems I: Higher integrability - Scheven, Christoph; Schmidt, Thomas (Journal of Differential Equations, 2010)
We consider weak solutions u of non-linear systems of partial differential equations. Assuming that the system exhibits a certain kind of elliptic behavior near infinity we prove higher integrability results for the gradient Du. In particular, we establish Hölder continuity of u in low dimensions. Moreover, we obtain analogous results for vectorial minimizers of multi-dimensional variational integrals. Finally, we discuss an extension to minimizing sequences and applications to generalized minimizers. © 2009 Elsevier Inc. All rights reserved.
UnivIS-Import:2015-03-09:Pub.2010.nat.dma.lma2.asympt
Keywords: Calculus of variations; Calderon-Zygmund estimates; Higher integrability; Minimizing sequence; Quasilinear elliptic system

Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure - Scheven, Christoph; Schmidt, Thomas (Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 2009)
We consider multidimensional variational integrals for vector-valued functions u : ℝn ⊃ Ω → ℝN. Assuming that the integrand satisfies the standard smoothness, convexity and growth assumptions only near ∞ we investigate the partial regularity of minimizers (and generalized minimizers) u. Introducing the open set R(u) : = x ∈ Ω : u is Lipschitz near x], we prove that R(u) is dense in Ω, but we demonstrate for n ≥ 3 by an example that Ω R(u) may have positive measure. In contrast, for n = 2 one has R(u) = Ω. Additionally, we establish analogous results for weak solutions of quasilinear elliptic systems.
UnivIS-Import:2015-03-09:Pub.2009.nat.dma.lma2.asympt

BV Minimizers of Variational Integrals: Existence, Uniqueness, Regularity - Schmidt, Thomas (2015)

Compactness results for normal currents and the Plateau problem in dual Banach spaces - Ambrosio, Luigi; Schmidt, Thomas (Proceedings of the London Mathematical Society, 2013)
We consider the Plateau problem and the corresponding free boundary problem for finite-dimensional surfaces in possibly infinite-dimensional Banach spaces. For a large class of duals and, in particular, for reflexive spaces, we establish the general solvability of these problems in terms of currents. As an auxiliary result, we prove a new compactness theorem for currents in dual spaces, which in turn relies on a fine analysis of the w*-topology. © 2012 London Mathematical Society.
UnivIS-Import:2015-03-09:Pub.2013.nat.dma.lma2.compac

Convex duality and uniqueness for BV-minimizers - Beck, Lisa; Schmidt, Thomas (Journal of Functional Analysis, 2015)
UnivIS-Import:2015-03-09:Pub.2014.nat.dma.lma2.convex

Harmonische Approximation in der Regularitätstheorie - Schmidt, Thomas (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2004)

Interior gradient regularity for BV minimizers of singular variational problems - Beck, Lisa; Schmidt, Thomas (Nonlinear Analysis - Theory Methods & Applications, 2015)
UnivIS-Import:2015-03-09:Pub.2015.nat.dma.lma2.interi

Local and asymptotic regularity results for quasiconvex and quasimonotone problems - Carozza, Menita; Passarelli di Napoli, Antonia; Schmidt, Thomas; Verde, Anna (Quarterly Journal of Mathematics, 2012)
Considering vectorial integrals in the multidimensional calculus of variations and quasilinear elliptic systems of partial differential equations, we prove gradient regularity of minimizers and weak solutions, respectively. In contrast to the classical theory, we impose our assumptions on the structure functions only locally (i.e. near a single point) or asymptotically (i.e. near infinity). In particular, we point out relations between the local and the asymptotic point of view, and we discuss notions of quasiconvexity at infinity and quasimonotonicity at infinity, which arise in this context. © 2010 Published by Oxford University Press. All rights reserved.
UnivIS-Import:2015-03-09:Pub.2012.nat.dma.lma2.locala

On the Dirichlet problem for variational integrals in BV - Beck, Lisa; Schmidt, Thomas (Journal für die reine und angewandte Mathematik, 2013)
We investigate the Dirichlet problem for multidimensional variational integrals with linear growth which is formulated in a generalized way in the space of functions of bounded variation. We prove uniqueness of minimizers up to additive constants and deduce additional assertions about these constants and the possible (non-)attainment of the boundary values. Moreover, we provide several related examples. In the case of the model integral [equation presented] our results extend classical results from the scalar case N = 1-where the problem coincides with the non-parametric least area problem-to the general vectorial setting N N. © 2013 Walter de Gruyter Berlin-Boston.
UnivIS-Import:2015-03-09:Pub.2013.nat.dma.lma2.onthed

On the Dirichlet problem for variational integrals in BV - Schmidt, Thomas (Partial Differential Equations, 2011)

Partial regularity for degenerate variational problems and image restoration models in BV - Schmidt, Thomas (Indiana University Mathematics Journal, 2014)
We establish partial and local C1,α-regularity results for vectorial almost-minimizers of convex variational integrals in BV. In particular, we investigate cases with a degenerate or singular behavior of p-Laplace type, and we cover (local) minimizers of the exemplary integrals ∫ Ω (1 + |∇w|p)1/p dχ with 1 < p < ∞. We also treat some related models with lower-order terms, which are motivated by image restoration. © Indiana University Mathematics Journal.
UnivIS-Import:2015-03-09:Pub.2014.nat.dma.lma2.partia

Partial regularity for mass-minimizing currents in Hilbert spaces - Ambrosio, Luigi; De Lellis, Camillo; Schmidt, Thomas (Journal für die reine und angewandte Mathematik, 2015)
UnivIS-Import:2015-03-09:Pub.2015.nat.dma.lma2.partia

Partial regularity of strong local minimizers of quasiconvex integrals with (p,q)-growth - Schemm, Sabine; Schmidt, Thomas (Proceedings of the Royal Society of Edinburgh Section A-Mathematics, 2009)
We consider strictly quasiconvex integrals in the multi-dimensional calculus of variations. For the C2-integrand f : Nn we impose (p, q)-growth conditions with , > 0 and 1 < p q < min p + 1/n, p(2n 1)/(2n 2). Under these assumptions we prove partial C1,loc- regularity for strong local minimizers of F and the associated relaxed functional F. © 2009 Copyright Royal Society of Edinburgh.
UnivIS-Import:2015-03-09:Pub.2009.nat.dma.lma2.partia_3

Regularity of minimizers of W¹·ᵖ-quasiconvex variational integrals with (p,q)-growth - Schmidt, Thomas (Calculus of Variations and Partial Differential Equations, 2008)
UnivIS-Import:2015-03-09:Pub.2008.nat.dma.lma2.regula

Regularity of relaxed minimizers of quasiconvex variational integrals with (p,q)-growth - Schmidt, Thomas (Archive for Rational Mechanics and Analysis, 2009)

Regularity theorems for degenerate quasiconvex energies with (p,q)-growth - Schmidt, Thomas (Advances in Calculus of Variations, 2008)
We study autonomous integrals F[u]:=∫ Ω f(Du)dx for u: ℝ n ⊃ Ω.→ ℝ N in the multidimensional calculus of variations, where the integrand f is a strictly quasiconvex function satisfying the (p, q)-growth conditions γ|ξ| p ≤ f(ξ) ≤ Γ(1 + |ξ| q) with exponents 1< p≤ q < p + min2, p/2n, Imposing the additional assumption that f resembles the degenerate behavior of the p-energy density, we establish a partial C 1,α-regularity theorem for F-minimizers and a similar theorem for minimizers of a relaxed functional. Our results cover the model case of polyconvex integrands f(ξ): = 1/p|ξ| p + h(det ξ), where h is a smooth convex function with q/n-growth. © de Gruyter 2008.
UnivIS-Import:2015-03-09:Pub.2008.nat.dma.lma2.regula_1
Keywords: Calculus of variations; Degeneration; Nonstandard growth; Partial regularity; Polyconvexity; Quasiconvexity; Relaxation

Strict interior approximation of sets of finite perimeter and functions of bounded variation - Schmidt, Thomas (Proceedings of the American Mathematical Society, 2015)
UnivIS-Import:2015-03-09:Pub.2014.nat.dma.lma2.strict

W²·¹⁺ᵋ estimates for the Monge-Ampère equation - Schmidt, Thomas (Advances in Mathematics, 2013)
UnivIS-Import:2015-03-09:Pub.2013.nat.dma.lma2.w21eps

Zur Existenz und Regularität von Minimierern quasikonvexer Variationsintegrale mit (p,q)-Wachstum - Schmidt, Thomas (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2006)