arXiv subjects

Luca Scarpa

Publications and source records attributed to Luca Scarpa.

59 records · Page 4Linked to original sources

arXiv2017

On the well-posedness of SPDEs with singular drift in divergence form

We prove existence and uniqueness of strong solutions for a class of second-order stochastic PDEs with multiplicative Wiener noise and drift of the form $\operatorname{div} γ(\nabla \cdot)$, where $γ$ is a maximal monotone graph in $\mathbb{R}^n \times \mathbb{R}^n$ obtained as the subdifferential of a convex function satisfying very mild assumptions on its behavior at infinity. The well-posedness result complements the corresponding one in our recent work arXiv:1612.08260 where, under the additional assumption that $γ$ is single-valued, a solution with better integrability and regularity properties is constructed. The proof given here, however, is self-contained.

math.AP↗

arXiv2016

Well-posedness for a class of doubly nonlinear stochastic PDEs of divergence type

We prove well-posedness for doubly nonlinear parabolic stochastic partial differential equations of the form $dX_t-\text{div}\,γ(\nabla X_t)\,dt+β(X_t)\,dt\ni B(t,X_t)\,dW_t$, where $γ$ and $β$ are the two nonlinearities, assumed to be multivalued maximal monotone operators everywhere defined on $\mathbb{R}^d$ and $\mathbb{R}$ respectively, and $W$ is a cylindrical Wiener process. Using variational techniques, suitable uniform estimates (both pathwise and in expectation) and some compactness results, well-posedness is proved under the classical Leray-Lions conditions on $γ$ and with no restrictive smoothness or growth assumptions on $β$. The operator $B$ is assumed to be Hilbert-Schmidt and to satisfy some classical Lipschitz conditions in the second variable.

math.AP↗

arXiv2016

From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation

A rigorous proof is given for the convergence of the solutions of a viscous Cahn-Hilliard system to the solution of the regularized version of the forward-backward parabolic equation, as the coefficient of the diffusive term goes to 0. Non-homogenous Neumann boundary condition are handled for the chemical potential and the subdifferential of a possible non-smooth double-well functional is considered in the equation. An error estimate for the difference of solutions is also proved in a suitable norm and with a specified rate of convergence.

math.AP↗

arXiv2015

Existence of solutions for a model of microwave heating

This paper is concerned with a system of differential equations related to a circuit model for microwave heating, complemented by suitable initial and boundary conditions. A RCL circuit with a thermistor is representing the microwave heating process with temperature-induced modulations on the electric field. The unknowns of the PDE system are the absolute temperature in the body, the voltage across the capacitor and the electrostatic potential. Using techniques based on monotonicity arguments and sharp estimates, we can prove the existence of a weak solution to the initial-boundary value problem.

math.AP↗

arXiv2014

A doubly nonlinear evolution problem related to a model for microwave heating

This paper is concerned with the existence and uniqueness of the solution to a doubly nonlinear parabolic problem which arises directly from a circuit model of microwave heating. Beyond the relevance from a physical point of view, the problem is very interesting also in a mathematical approach: in fact, it consists of a nonlinear partial differential equation with a further nonlinearity in the boundary condition. Actually, we are going to prove a general result: the two nonlinearities are allowed to be maximal monotone operators and then an existence result will be shown for the resulting problem.

math.AP↗