Related papers: A non-autonomous variational problem describing a …
We prove some regularity results for a priori bounded local minimizers of non-autonomous integral functionals of the form $$\mathcal{F}(v,\Omega)=\int_\Omega F(x,Dv)dx,$$ under the constraint $v \ge \psi$ a.e. in $\Omega$, where $\psi$ is a…
In this paper, we consider the following variational problem: \begin{eqnarray*} \inf_{u\in…
We give a simple proof of existence of solutions of the dispersion manage- ment and diffraction management equations for zero average dispersion, respectively diffraction. These solutions are found as maximizers of non-linear and non-local…
In this paper we study an equation driven by a nonlocal anisotropic operator with homogeneous Dirichlet boundary conditions. We find at least three non trivial solutions: one positive, one negative and one of unknown sign, using variational…
We consider the "thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in $\mathbb R^{n+1}_+$ plus the area of the positivity set of that function in $\mathbb R^n$. We establish full…
We study the non-linear minimization problem on $H^1_0(\Omega)\subset L^q$ with $q=\frac{2n}{n-2}$, $\alpha>0$ and $n\geq4$~: \[\inf_{\substack{u\in H^1_0(\Omega) \|u\|_{L^q}=1}}\int_\Omega a(x,u)|\nabla u|^2 - \lambda \int_{\Omega}…
In this paper, we develop a theoretical framework for nonlinear stochastic optimal control problems with optimal stopping by establishing a density-based deterministic representation of the underlying diffusion. For state-independent…
The thin obstacle problem or $n$-dimensional Signorini problem is a classical variational problem arising in several applications, starting with its first introduction in elasticity theory. The vast literature concerns mostly quadratic…
Electrified liquids are well known to be prone to a variety of interfacial instabilities that result in the onset of apparent interfacial singularities and liquid fragmentation. In the case of electrically conducting liquids, one of the…
Our main contributions include proving sufficient conditions for the existence of solution to a second order problem with nonzero nonlocal initial conditions, and providing a comprehensive analysis using fundamental solutions and…
A well-posed stress-driven mixture is proposed for Timoshenko nano-beams. The model is a convex combination of local and nonlocal phases and circumvents some problems of ill-posedness emerged in strain-driven Eringen-like formulations for…
We survey methods and results of fractional differential equations in which an unknown function is under the operation of integration and/or differentiation of fractional order. As an illustrative example, we review results on fractional…
We show that, for a fixed order $\gamma\geq 1$, each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to stationarity of order $1$),…
We consider the connection problem of the second nonlinear differential equation \begin{equation} \label{eq:1} \Phi''(x)=(\Phi'^2(x)-1)\cot\Phi(x)+ \frac{1}{x}(1-\Phi'(x)) \end{equation} subject to the boundary condition…
Motivated by the modeling of the temporal structure of the velocity field in a highly turbulent flow, we propose and study a linear stochastic differential equation that involves the ingredients of a Ornstein-Uhlenbeck process, supplemented…
We study a biologically inspired, inherently non-equilibrium model consisting of self-propelled particles. In the model, particles move on a plane with a velocity of constant magnitude; they locally interact with their neighbors by choosing…
We establish the existence and symmetry of all minimizers of a constrained variational problem involving the fractional gradient. This problem is closely connected to some fractional kinetic equations.
In this paper we study second order stochastic differential equations with measurable and density-distribution dependent coefficients. Through establishing a maximum principle for kinetic Fokker-Planck-Kolmogorov equations with…
In this paper, we study the nonlinear Vlasov-Fokker-Planck equation with fixed collision frequency. We establish the global-in-time existence of weak solutions to the equation with large initial data. Moreover, we show that our solution…
We use the Wa\'zewski topological principle to establish a number of new sufficient conditions for the existence of proper (defined on the entire time axis) solutions of essentially nonlinear nonautonomous systems. The systems under…