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Stefania PERROTTA

Ricercatore Universitario
Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Matematica

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Pubblicazioni

2024 - Evolution equations with nonlocal multivalued Cauchy problems [Articolo su rivista]
Malaguti, L.; Perrotta, S.
abstract

We consider evolution equations in Banach spaces. Their linear parts generate a strongly continuous C0-semigroup of contractions. The nonlinear term is a Carathéodory function. When the semigroup is not compact the nonlinearity has an additional restriction, involving the Hausdorff measure of noncompactness. We provide solutions satisfying nonlocal, multivalued Cauchy conditions. Our approach involves a suitable degree argument. The duality mapping is used for guaranteeing the lack of fixed points of the associated homotopic fields along the boundary of their domain. We apply our results for the investigation of transport and diffusion equations for which we provide the existence of nonlocal solutions.

2022 - Local Lipschitz continuity for energy integrals with slow growth [Articolo su rivista]
Eleuteri, M.; Marcellini, P.; Mascolo, E.; Perrotta, S.
abstract

We consider some energy integrals under slow growth, and we prove that the local minimizers are locally Lipschitz continuous. Many examples are given, either with subquadratic p, q- growth and/or anisotropic growth.

2022 - Lp-exact controllability of partial differential equations with nonlocal terms [Articolo su rivista]
Malaguti, Luisa; Perrotta, Stefania; Taddei, Valentina
abstract

The paper deals with the exact controllability of partial differential equations by linear controls. The discussion takes place in infinite dimensional state spaces since these equations are considered in their abstract formulation as semilinear equations. The linear parts are densely defined and generate strongly continuous semigroups. The nonlinear terms may also include a nonlocal part. The solutions satisfy nonlocal properties, which are possibly nonlinear. The states belong to Banach spaces with a Schauder basis and the results exploit topological methods. The novelty of this investigation is in the use of an approximation solvability method which involves a sequence of controllability problems in finite-dimensional spaces. The exact controllability of nonlocal solutions can be proved, with controls in Lp spaces, 1<∞. The results apply to the study of the exact controllability for the transport equation in arbitrary Euclidean spaces and for the equation of the nonlinear wave equation.

2019 - Exact controllability of infinite dimensional systems with controls of minimal norm [Articolo su rivista]
Malaguti, Luisa; Perrotta, Stefania; Taddei, Valentina
abstract

The paper deals with the exact controllability of a semilinear system in a separable Hilbert space. A bounded linear part is considered and a linear control introduced. The state space is compactly embedded in a Banach space and the nonlinear term is continuous in its state variable in the norm of the Banach space. An infinite sequence of finite dimen- sional controllability problems is introduced and the solution is obtained by a limiting procedure. To the best of our knowledge, the method is new in controllability theory. An application to an integro-differential system in euclidean spaces completes the discussion.

2013 - Polyconvex energies and cavitation [Articolo su rivista]
P., Celada; Perrotta, Stefania
abstract

We study the existence of singular minimizers in the class of radial deformations for polyconvex energies that grow linearly with respect to the Jacobian.

2008 - A correction of the paper "On minima of radially symmetric functionals of the gradient" [Articolo su rivista]
A., Cellina; Perrotta, Stefania
abstract

We prove a theorem for the existence of solutions to a variational problem, under assumptions that do not require the convexity of the integrand.

2007 - On a class of nonconvex Bolza problems related to Blatz-Ko elastic materials [Articolo su rivista]
P., Celada; Perrotta, Stefania
abstract

We study the existence of solutions to Bolza problems for a special class of one-dimensional, nonconvex integrals. These integrals describe the possibly singular, radial deformations of certain rubberlike materials called Blatz–Ko materials.

2002 - Existence of minimizers for nonconvex, noncoercive simple integrals. [Articolo su rivista]
P., Celada; Perrotta, Stefania
abstract

We consider the problem of minimizing autonomous, simple integrals such as \min\,\left\{ \int_0^T f\left(x(t)\,,x^\prime(t)\right)\,dt\colon\,\, \text{$x\in AC{([0\,,T])}$, $x(0)=x_0$, $x(T)=x_T$} \right\}, \tag{$\cal{P}$} where $f:{\mathbb R}\times{\mathbb R} \to [0,\infty]$ is a possibly nonconvex function with either superlinear or slow growth at infinity. Assuming that the relaxed problem ($\cal{P}^{\ast\ast}$)---obtained from ($\cal{P}$) by replacing f with its convex envelope f** with respect to the derivative variable $x^\prime$---admits a solution, we prove attainment for ($\cal{P}$) under mild regularity and growth assumptions on f and f**. We discuss various instances of growth conditions on f that yield solutions to the corresponding relaxed problem ($\cal{P}^{\ast\ast}$), and we present examples that show that the hypotheses on f and f** considered here for attainment are essentially sharp.

2001 - Minimizing nonconvex, simple integrals of product type [Articolo su rivista]
P., Celada; Perrotta, Stefania
abstract

We consider the problem of minimizing simple integrals of product type, i.e. min {integral (T)(0) g(x(t))f(x ´ (t)) dt: x is an element of AC([0, T]), x(0) = x(0), x(T) = x(T)}. where f:R --> [0, proportional to] is a possibly nonconvex, lower semicontinuous function with either superlinear or slow growth at infinity. Assuming that the relaxed problem (P**) obtained from (P) by replacing f with its convex envelope f** admits a solution. we prove attainment for (P) for every continuous, positively bounded below the coefficient g such that (i) every point t is an element ofR is squeezed between two intervals where g is monotone and (ii) g has no strict local minima. This shows in particular that, for those f such that the relaxed problem (P**) has a solution, the class of coefficients g that yield existence to (P) is dense in the space of continuous, positive Functions on R. We discuss various instances of growth conditions on f that yield solutions to (P**) and we present examples that show that the hypotheses on g considered above for attainment are essentially sharp.

2001 - On the minimum problem for nonconvex, multiple integrals of product type [Articolo su rivista]
P., Celada; Perrotta, Stefania
abstract

We consider the problem of minimizing multiple integrals of product type, i.e. (P) min [GRAPHICS] where Omega is a bounded, open set in R-N, f: R-N --> [0, infinity) is a possibly nonconvex, lower semicontinuous function with p-growth at infinity for some 1 < p < infinity and the boundary datum u(0) is in W-1,W-p(Omega) boolean AND L-infinity(Omega) (or simply in W-1,W-p(Omega) if N < p < infinity). Assuming that the convex envelope f** of f is affine on each connected component of the set {f** < f}, we prove attainment for (P) for every continuous, positively bounded below function g such that (i) every point t <is an element of> R is squeezed between two intervals where g is monotone and (ii) g has no strict local minima. This shows in particular that the class of coefficents g that yield existence to (P) is dense in the space of continuous, positive functions on R. We present examples which show that these conditions for attainment are essentially sharp.

2000 - Minimizing non-convex multiple integrals: a density result. [Articolo su rivista]
P., Celada; Perrotta, Stefania
abstract

We consider variational problems whose lagrangian is of the form f(Du)+g(u) where f is a possibly non-convex lower semicontinuous function with p-growth at infinity for some 1 < p < ∞, and the boundary datum is any function in W 1,p (Ω). Assuming that the convex envelope of f is affine on each connected component of the set {f ^∗∗ < f }, we prove the existence of solutions to (P) for every continuous function g such that (i) g has no strict local minima and (ii) every convergent sequence of extremum points of g eventually belongs to an interval where g is constant, thus showing that the set of continuous functions g that yield existence to (P) is dense in the space of continuous functions on R.

2000 - Vectorial Hamilton-Jacobi equations with rank one affine dependence on the gradient. [Articolo su rivista]
P., Celada; Perrotta, Stefania
abstract

This paper deals with Dirichlet problems for vectorial, stationary Hamilton-Jacobi equations

1999 - Nonconvex variational problems related to a hyperbolic equation [Articolo su rivista]
F., FLORES BAZAN; Perrotta, Stefania
abstract

We first prove a new Lyapunov-type theorem which will yield existence of solutions to nonconvex minimum problems involving some hyperbolic equations on rectangular domains with Darboux boundary conditions. Some problems with obstacle and bang-bang results are also considered.

1998 - Existence of solutions for a class of non convex minimum problems [Articolo su rivista]
P., Celada; Perrotta, Stefania; G., Treu
abstract

In this paper we give sufficient conditions for the existence of solutions to the problem of minimizing the integral of [f ( ∇v) + v] on a convex n-dimensional set Ω . Here f is nonnegative, nonconvex, Borel-measurable, and vanishes on the boundary of a convex n-dimensional set K.

1998 - Functions with prescribed singular values of the gradient. [Articolo su rivista]
P., Celada; Perrotta, Stefania
abstract

We prove the existence of infinitely many vector-valued Lipschitz-continuous functions u on an open set Ω satisfying suitable Dirichlet boundary conditions such that the singular values of the gradient matrix ∇u, agree a.e. on Ω with N given positive, bounded and lower semicontinuous functions.

1998 - On the validity of the maximum principle and of the Euler-Lagrange equation for a minimum problem depending on the gradient [Articolo su rivista]
A., Cellina; Perrotta, Stefania
abstract

We consider the limiting case alpha = infinity of the problem of minimizing integral(Omega) (\\del u(x)\\(alpha) + g(u))dx on u is an element of + u(0) + W-0(1, alpha) (Omega); where g is differentiable and strictly monotone. If this infimum is finite, it is evidently attained; we show that any minimizing function u satisfies the appropriate form of the Euler-Lagrange equation, i.e., for some function p, div p(x) = g'(u(x)) for p(x) is an element of partial derivative(jB)(del(x)); where j(B) is the indicator function of the closed unit ball in the Euclidean norm of R-N and partial derivative is the subdifferential of the convex function j(B).

1995 - On a problem of potential wells. [Articolo su rivista]
A., Cellina; Perrotta, Stefania
abstract

We find an explicit solution for a potential wells problem in dimension 3.

1994 - On minima of radially symmetric functionals of the gradient. [Articolo su rivista]
A., Cellina; Perrotta, Stefania
abstract

In this paper we consider the problems of the existence, the uniqueness and the qualitative properties (symmetry) of the minima to a minimization problem in the calculus of variations.

1994 - On the closure of reachable sets for control systems. [Articolo su rivista]
Perrotta, Stefania
abstract

We prove a density result related to control systems with closed reachable set.