Paolo Marcellini

Regularity and existence of solutions of elliptic equations with p,q-growth conditions

for some positive constants m, A4, and for exponents q 2 p 2 2. Under (1.2), (1.3), and some other assumptions, by assuming also that the quotient q/p is sufficiently close to one in dependence on n (precisely, if q/p c n/(n 2)), then we prove that every weak solution to (1.1) of class ?4’:;,4(0) is locally Lipschitz-continuous in Q. Moreover, there are positive constants 8, c, and 0 2 1 such that

Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals

AbstractWe study semicontinuity of multiple integrals ∫Ωf(x,u,Du) dx, where the vector-valued function u is defined for

General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases

x \varepsilon \Omega \subset \mathbb{R}^n

Do ungulates exhibit a food density threshold ? A field study of optimal foraging and movement patterns

with values in ℝN. The function f(x,s,ξ) is assumed to be Carathéodory and quasiconvex in Morreys sense. We give conditions on the growth of f that guarantee the sequential lower semicontinuity of the given integral in the weak topology of the Sobolev space H1,p(ΩℝN). The proofs are based on some approximation results for f. In particular we can approximate f by a nondecreasing sequence of quasiconvex functions, each of them beingconvex andindependent of (x,s) for large values of ξ. In the special polyconvex case, for example if n=N and f(Du) is equal to a convex function of the Jacobian detDu, then we obtain semicontinuity in the weak topology of H1,p(Ωℝn) for small p, in particular for some p smaller than n.

Semicontinuity problems in the calculus of variations

SummaryWe prove an homogenization formula for some non linear variational problems that extends the analogous one known in the linear case. Namely the solution uε of the problem

Relaxation of Multiple Integrals in Subcritical Sobolev Spaces

\int\limits_\Omega {\left\{ {f\left( {\frac{x}{\varepsilon },Du_\varepsilon } \right) - \varphi \left( x \right)u_\varepsilon } \right\}dx} = minimum,