Volume 12, Issue 1

We consider an evolution in phase-field fracture which combines, in a system of PDEs, an irreversible gradient-flow for the phase-field variable with the equilibrium equation for the displacement field. We introduce a discretization in time and define a discrete solution by means of a 1-step alternate minimization scheme, with a quadratic L2{L^{2}}-penalty in the phase-field variable (i.e. an alternate minimizing movement). First, we prove that discrete solutions converge to a solution of our system of PDEs. Then we show that the vanishing viscosity limit is a quasi-static (parametrized) BV-evolution. All these solutions are described both in terms of energy balance and, equivalently, by PDEs within the natural framework of W1,2⁢(0,T;L2){W^{1,2}(0,T;L^{2})}.

We consider a parametric nonlinear elliptic equation driven by the Robin p -Laplacian. The reaction term is a Carathéodory function which exhibits competing nonlinearities (concave and convex terms). We prove two bifurcation-type results describing the set of positive solutions as the parameter varies. In the process we also prove two strong comparison principles for Robin equations. These results are proved for differential operators which are more general than the p -Laplacian and need not be homogeneous.

We study integrals of the form ∫Ωf⁢(d⁢ω1,…,d⁢ωm){\int_{\Omega}f(d\omega_{1},\dots,d\omega_{m})}, where m≥1{m\geq 1} is a given integer, 1≤ki≤n{1\leq k_{i}\leq n} are integers, ωi{\omega_{i}} is a (ki-1){(k_{i}-1)}-form for all 1≤i≤m{1\leq i\leq m} and f:∏i=1mΛki⁢(ℝn)→ℝ{f:\prod_{i=1}^{m}\Lambda^{k_{i}}(\mathbb{R}^{n})\rightarrow\mathbb{R}} is a continuous function. We introduce the appropriate notions of convexity, namely vectorial ext. one convexity, vectorial ext. quasiconvexity and vectorial ext. polyconvexity. We prove weak lower semicontinuity theorems and weak continuity theorems and conclude with applications to minimization problems. These results generalize the corresponding results in both classical vectorial calculus of variations and the calculus of variations for a single differential form.

We prove the higher differentiability and the higher integrability of the a priori bounded local minimizers of integral functionals of the form ℱ⁢(v,Ω)=∫Ωf⁢(x,D⁢v⁢(x))⁢dx,\mathcal{F}(v,\Omega)=\int_{\Omega}f(x,Dv(x))\,{\mathrm{d}}x, with convex integrand satisfying p -growth conditions with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x -variable belongs to a suitable Sobolev space. The a priori boundedness of the minimizers allows us to obtain the higher differentiability under a Sobolev assumption which is independent on the dimension n and that, in the case p≤n-2{p\leq n-2}, improves previous known results. We also deal with solutions of elliptic systems with discontinuous coefficients under the so-called Uhlenbeck structure. In this case, it is well known that the solutions are locally bounded and therefore we obtain analogous regularity results without the a priori boundedness assumption.