1,720,980 research outputs found
An elementary approach to the dimension of measures satisfying a first-order linear PDE constraint
No full textWe give a simple criterion on the set of probability tangent measures 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
A Lebesgue-Lusin property for linear operators of first and second order
No full textWe prove that for a homogeneous linear partial differential operator A of order k <= 2 and an integrable map f taking values in the essential range of that operator, there exists a function u of special bounded variation satisfying Au(x) = f (x) almost everywhere.This extends a result of G. Alberti for gradients on R-N. In particular, for 0 <= m < N, it is shown that every integrable m-vector field is the absolutely continuous part of the boundary of a normal (m + 1)-current
Regularity for free interface variational problems in a general class of gradients
No full textWe 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 (Formula presented.) ,where Ω is a bounded Lipschitz domain, A⊂ RN is a Borel set, u: Ω ⊂ RN→ Rd, A is an operator of gradient form, and σ1, σ2 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∩ Ω is locally a C 1-hypersurface up to a closed set of zero HN-1-measure
Characterization of Generalized Young Measures Generated by A-free Measures
Full text linkWe give two characterizations, one for the class of generalized Young measures generated by A-free measures and one for the class generated by B-gradient measures Bu. Here, A and B are linear homogeneous operators of arbitrary order, which we assume satisfy the constant rank property. The first characterization places the class of generalized A-free Young measures in duality with the class of A-quasiconvex integrands by means of a well-known Hahn–Banach separation property. The second characterization establishes a similar statement for generalized B-gradient Young measures. Concerning applications, we discuss several examples that showcase the failure of L 1-compensated compactness when concentration of mass is allowed. These include the failure of L 1-estimates for elliptic systems and the lack of rigidity for a version of the two-state problem. As a byproduct of our techniques we also show that, for any bounded open set Ω , the inclusions L1(Ω)∩kerA↪M(Ω)∩kerA,{Bu∈C∞(Ω)}↪{Bu∈M(Ω)}are dense with respect to the area-functional convergence of measures
Generalized Multiscale Young Measures
No full textThis paper is devoted to the construction of generalized multiscale Young measures, which are the extension of Pedregal's multiscale Young measures [Trans. Amer. Math. Soc., 358 (2006), pp. 591-602] to the setting of generalized Young measures introduced by DiPerna and Majda [Comm. Math. Phys., 108 (1987), pp. 667-689]. As a tool for variational problems, these are well-suited objects for the study (at different length-scales) of oscillation and concentration effects of convergent sequences of measures. Important properties of multiscale Young measures such as compactness, representation of nonlinear compositions, localization principles, and differential constraints are extensively developed in the second part of this paper. As an application, we use this framework to address the Gamma-limit characterization of the homogenized limit of convex integrals defined on spaces of measures satisfying a general linear PDE-constraint
Relaxation and optimization for linear-growth convex integral functionals under PDE constraints
No full textWe give necessary and sufficient conditions for the minimality of generalized minimizers of linear-growth integral functionals of the form F[u]=∫Ωf(x,u(x))dx,u:Ω⊂Rd→RN, where f:Ω×RN→R is a convex integrand and 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 M(Ω;RN), and the introduction of a set-valued pairing on M(Ω;RN)×L∞(Ω;RN). 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 and divergence-free spaces
A Bourgain-Brezis-Mironescu representation for functions with bounded deformation
Full text linkWe establish a non-local integral difference quotient representation for symmetric gradient semi-norms in BD(Ω) and LD(Ω), which does not require the manipulation of distributional derivatives. Our representation extends the formulas for the symmetric gradient established by Mengesha for vector-fields in W1,p(Ω;Rd), which are inspired by the gradient semi-norm formulas introduced by Bourgain, Brezis and Mironescu in W1,p(Ω) and by Dávila in BV(Ω)
An elementary approach to the homological properties of constant-rank operators
Full text linkWe give a simple and constructive extension of Raiță’s result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement over the order of the operators constructed by Raiță and the order of the explicit annihilators for elliptic operators due to Van Schaftingen. We also give an abstract construction of an optimal annihilator for constant-rank operators, which extends the optimal construction of Van Schaftingen for elliptic operators. Lastly, we discuss the homological properties of operators in relation to the homological properties of their associated symbols. We establish that the constant-rank property is a sufficient and necessary condition for the validity of a generalized Poincaré lemma on spaces of homogeneous maps over , and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property
Dimensional estimates and rectifiability for measures satisfying linear PDE constraints
No full textWe establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. In particular, our general theorem provides a new proof of the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD). For divergence-free tensors we obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures
Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints
Full text linkWe 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
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