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By exploiting a suitable Trudinger-Moser inequality for fractional Sobolev spaces, we obtain existence and multiplicity of solutions for a class of one-dimensional nonlocal equations with fractional diffusion and nonlinearity at exponential…
We prove that the Cauchy problem associated to the Zakharov-Schulman system $iu_t+L_1u=uv$, $L_2v=L_3(|u|^2)$ is locally well-posed for given initial data in Sobolev spaces $H^s(R^n)$, $s\geq n/4$, for n =2,3. Here, L_j denote second order…
This monograph is centred at the intersection of three mathematical topics, that are theoretical in nature, yet with motivations and relevance deep rooted in applications: the linear inverse problems on abstract, in general…
We consider a slightly subcritical elliptic system with Dirichlet boundary conditions and a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of…
We prove the existence of local-in-time smooth solutions of the incompressible semi-geostrophic equations expressed in Eulerian co-ordinates in 3-dimensional smooth bounded simply-connected domains. Our solutions adhere to Cullen's…
In this paper we prove the existence of solutions to the cubic NLS equation with convolution potentials on two dimensional irrational tori undergoing an arbitrarily large growth of Sobolev norms as time evolves. Our results apply also to…
We consider the Shallow Water equations in the supercritical and subcritical cases in one space variable,posed in a finite spatial interval with characteristic boundary conditions at the endpoints, which, as is well known, are…
In this paper we study the numerical method for approximating the random periodic solution of semiliear stochastic evolution equations. The main challenge lies in proving a convergence over an infinite time horizon while simulating…
We prove local in time well-posedness in Sobolev spaces of the Cauchy problem for semi-linear p-evolution equations of the first order with real principal part, but complex valued coefficients for the lower order terms, assuming decay…
The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients…
We establish existence of positive non-decreasing radial solutions for a nonlocal nonlinear Neumann problem both in the ball and in the annulus. The nonlinearity that we consider is rather general, allowing for supercritical growth (in the…
We are concerned with Dirichlet problems of the form $${\mathop{\rm div}\nolimits} (|D u|^{p-2}Du)+f(u)=0\ \mbox{ in }\Omega,\qquad u=0\ \mbox{ on }\partial\Omega, $$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$, $n\ge 2$, $1<p<n$…
In this paper we study the positive solutions of sub linear elliptic equations with a Hardy potential which is singular at the boundary. By means of ODE techniques a fairly complete picture of the class of radial solutions is given. Local…
We are concerned with a semilinear elliptic equation in the half-space, subject to a nonlinear dynamic boundary condition. We establish the global well-posedness of solutions in a new setting for the problem, namely the framework of Morrey…
Euler equations are the basic system in fluid dynamics describing the motion of incompressible and inviscid ideal fluids. For a bounded smooth domain $\Omega$ in $\mathbb{R}^n$. The well-posedness of Euler equations is well-known in Sobolev…
The authors propose and analyze a well-posed numerical scheme for a type of ill-posed elliptic Cauchy problem by using a constrained minimization approach combined with the weak Galerkin finite element method. The resulting Euler-Lagrange…
The existence of solutions to some initial-boundary value problem for the Stokes system is proved. The result is shown in Sobolev-Slobodetskii spaces such that the velocity belongs to $W_r^{2+\sigma,1+\sigma/2}(\Omega^T)$ and gradient of…
This work deals with Schr\"odinger equations with quadratic and sub-quadratic Hamiltonians perturbed by a potential. In particular we shall focus on bounded, but not necessarily smooth perturbations. We shall give a representation of such…
An elementary treatment of the Dirac Equation in the presence of a three-dimensional spherically symmetric $\delta (r-r_0)$-potential is presented. We show how to handle the matching conditions in the configuration space, and discuss the…
We study a Sobolev-type inequality involving the $p$-curl operator in $\mathbb{R}^3$. We prove the existence of a minimizer which yields a solution to the $p$-curl-curl equation in the critical case. The problem is motivated both by…