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Adolfo Arroyo-Rabasa

Publications and source records attributed to Adolfo Arroyo-Rabasa.

23 records · Page 2Linked to original sources
arXiv2018

An elementary approach to the dimension of measures satisfying a first-order linear PDE constraint

We give a simple criterion on the set of probability tangent measures $\mathrm{Tan}(μ,x)$ of a positive Radon measure $μ$, which yields lower bounds on the Hausdorff dimension of $μ$. As an application, we give an elementary and purely algebraic proof of the sharp Hausdorff dimension lower bounds for first-order linear PDE-constrained measures; bounds for closed (measure) differential forms and normal currents are further discussed. A weak structure theorem in the spirit of [Ann. Math. 184(3) (2016), pp. 1017-1039] is also discussed for such measures.

math.AP↗
arXiv2017

Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities of measure solutions to linear PDEs and of the generalized convexity notions corresponding to these PDE constraints.

math.AP↗
arXiv2017

Relaxation and optimization for linear-growth convex integral functionals under PDE constraints

We give necessary and sufficient conditions for minimality of generalized minimizers for linear-growth functionals of the form \[ \mathcal F[u] := \int_Ωf(x,u(x)) \, \text{d}x, \qquad u:Ω\subset \mathbb R^N\to \mathbb R^d, \] where $u$ is an integrable function satisfying a general PDE constraint. Our analysis is based on two ideas: a relaxation argument into a subspace of the space of bounded vector-valued Radon measures $\mathcal M(Ω;\mathbb R^d)$, and the introduction of a set-valued pairing in $\mathcal M(Ω;\mathbb R^N) \times {\rm L}^\infty(Ω;\mathbb R^N)$. By these means we are able to show an intrinsic relation between minimizers of the relaxed problem and maximizers of its dual formulation also known as the saddle-point conditions. In particular, our results can be applied to relaxation and minimization problems in BV, BD.

math.AP↗
arXiv2016

Regularity for free interface variational problems in a general class of gradients

We present a way to study a wide class of optimal design problems with a perimeter penalization. More precisely, we address existence and regularity properties of saddle points of energies of the form $$ (u,A) \quad \mapsto \quad \int_Ω2fu \; \text{d}x \; - \int_{Ω\cap A} σ_1 \mathscr A u \cdot \mathscr A u \; \text{d}x \; - \int_{Ω\setminus A} σ_2\mathscr A u\cdot \mathscr A u \; \text{d}x \; + \; \text{Per}(A;\overline Ω),$$ where $Ω$ is a bounded Lipschitz domain, $A\subset \mathbb R^N$ is a Borel set, $u:Ω\subset \mathbb R^N \to \mathbb R^d$, $\mathscr A$ is an operator of gradient form, and $σ_1, σ_2$ are two not necessarily well-ordered symmetric tensors. The class of operators of gradient form includes scalar- and vector-valued gradients, symmetrized gradients, and higher order gradients. Therefore, our results may be applied to a wide range of problems in elasticity, conductivity or plasticity models. In this context and under mild assumptions on $f$, we show for a solution $(w,A)$, that the topological boundary of $A \cap Ω$ is locally a $\rm{C}^1$-hypersurface up to a closed set of zero $\mathscr H^{N-1}$-measure.

math.OC↗