Volume 13, Issue 3

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.

Let H be a norm of ℝN{\mathbb{R}^{N}} and H0{H_{0}} the dual norm of H . Denote by ΔH{\Delta_{H}} the Finsler–Laplace operator defined by ΔH⁢u:=div⁡(H⁢(∇⁡u)⁢∇ξ⁡H⁢(∇⁡u)){\Delta_{H}u:=\operatorname{div}(H(\nabla u)\nabla_{\xi}H(\nabla u))}. In this paper we prove that the Finsler–Laplace operator ΔH{\Delta_{H}} acts as a linear operator to H0{H_{0}}-radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution to the Cauchy problem for the Finsler heat equation ∂t⁡u=ΔH⁢u,x∈ℝN,t>0,\partial_{t}u=\Delta_{H}u,\quad x\in\mathbb{R}^{N},\,t>0, where N≥1{N\geq 1} and ∂t:=∂∂⁡t{\partial_{t}:=\frac{\partial}{\partial t}}.

Integrals of the Calculus of Variations with p,q{p,q}-growth may have not smooth minimizers, not even bounded, for general p,q{p,q} exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand f⁢(x,ξ){f(x,\xi)} with dependence on the modulus of the gradient, i.e. f⁢(x,ξ)=g⁢(x,|ξ|){f(x,\xi)=g(x,|\xi|)}. Without imposing structure conditions, we prove that if qp{\frac{q}{p}} is sufficiently close to 1, then every minimizer is locally Lipschitz-continuous.

The present paper arose from recent studies of energy-minimal deformations of planar domains. We are concerned with the Dirichlet energy. In general the minimal mappings need not be homeomorphisms. In fact, a part of the domain near its boundary may collapse into the boundary of the target domain. In mathematical models of nonlinear elasticity this is interpreted as interpenetration of matter . We call such occurrence the Nitsche phenomenon , after Nitsche’s remarkable conjecture (now a theorem) about existence of harmonic homeomorphisms between annuli. Indeed the round annuli proved to be perfect choices to grasp the nuances of the problem. Several papers are devoted to a study of deformations of annuli. Because of rotational symmetry it seems likely that the Dirichlet energy-minimal deformations are radial maps. That is why we confine ourselves to radial minimal mappings. The novelty lies in the presence of a weight in the Dirichlet integral. We observe the Nitsche phenomenon in this case as well, see our main results Theorem 1.4 and Theorem 1.7. However, the arguments require further considerations and new ingredients. One must overcome the inherent difficulties arising from discontinuity of the weight. The Lagrange–Euler equation is unavailable, because the outer variation violates the principle of none interpenetration of matter. Inner variation, on the other hand, leads to an equation that involves the derivative of the weight. But our weight function is only measurable which is the main challenge of the present paper.