Related papers: Non-trapping magnetic fields and Morrey-Campanato …
We present a rigorous mathematical treatment of the zero-field orbital magnetic susceptibility of a non-interacting Bloch electron gas, at fixed temperature and density, for both metals and semiconductors/insulators. In particular, we…
We consider the semilinear electromagnetic Schr\"{o}dinger equation (-i\nabla+A(x))^{2}u + V(x)u = |u|^{2^{\ast}-2}u, u\in D_{A,0}^{1,2}(\Omega,\mathbb{C}), where $\Omega=(\mathbb{R}^{m}\smallsetminus{0})\times\mathbb{R}^{N-m}$ with $2\leq…
We present a simple construction of a random Schr\"odinger operator subject to a magnetic field with a regularity as low as $0^-$-H\"older and a Gaussian white noise electric potential on a two-dimensional bounded box. This construction is…
The nonlinear selfdual variational principle established in a preceeding paper [8] -- though good enough to be readily applicable in many stationary nonlinear partial differential equations -- did not however cover the case of nonlinear…
The work treats smoothing and dispersive properties of solutions to the Schrodinger equation with magnetic potential. Under suitable smallness assumption on the potential involving scale invariant norms we prove smoothing - Strichartz…
In this manuscript, we employ the Nash-Moser iteration technique to determine a condition under which the positive solution $u$ of the generalized nonlinear Poisson equation $$\operatorname{div} (\varphi(|\nabla u|^2)\nabla u) + \psi(u^2)u…
In $L_2 (\mathbb{R}^d; \mathbb{C}^n)$, we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{A}_\varepsilon$ with periodic coefficients depending on $\mathbf{x}/\varepsilon$. We find approximations…
We compute the magnetic response of hollow semimetal cylinders and rings to the presence of an axial Aharonov-Bohm magnetic flux, in the absence of interactions. We predict nullification of the Aharonov-Bohm effect for a class of dispersion…
We prove cyclotron damping for the collisionless Vlasov-Maxwell equations on $\mathbb{T}_{x}^{3}\times\mathbb{R}_{v}^{3}$ under the assumptions that the electric induction is zero and $(\mathcal{\mathbf{PSC}})$ holds. It is a crucial step…
We investigate the general plasma-vacuum interface problems for the ideal incompressible MHD equations with or without surface tension and prove their nonlinear local well-posedness in standard Sobolev spaces under either non-zero surface…
In this paper we prove a nonexistence result for nonlinear parabolic problems with zero lower order term whose model is $$ \begin{cases} u_{t}- \Delta_p u+|u|^{q-1}u=\lambda & \text{in}\ (0,T)\times\Omega u(0,x)=0 & \text{in}\ \Omega,\\…
This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces,…
We prove a weak Harnack estimate for a class of doubly nonlinear nonlocal equations modelled on the nonlocal Trudinger equation \begin{align*} \partial_t(|u|^{p-2}u) + (-\Delta_p)^s u = 0 \end{align*} for $p\in (1,\infty)$ and $s \in…
We study the qualitative properties of groundstates of the time-independent magnetic semilinear Schr\"odinger equation \[ - (\nabla + i A)^2 u + u = |u|^{p-2} u, \qquad \text{ in } \mathbb{R}^N, \] where the magnetic potential $A$ induces a…
We study magnetic traps with very high trap frequencies where the spin is coupled to the motion of the atom. This allows us to investigate how the Born-Oppenheimer approximation fails and how effective magnetic and electric fields appear as…
In this paper, we are concerned with the estimates for the moments of stochastic convolution integrals. We first deal with the stochastic singular integral operators and we aim to derive the Morrey-Campanato estimates for the $p$-moments…
We prove global-in-time Strichartz estimates without loss of derivatives for the solution of the Schroedinger equation on a class of non-trapping asymptotically conic manifolds. We obtain estimates for the full set of admissible indices,…
In this paper, combining Nash-Moser iteration and Sallof-Coste type Sobolev ineualities, we establish fundamental and concise $C^0$ and $C^1$ estimates for solutions to a class of nonlinear elliptic equations of the form $$\Delta…
We examine semiclassical magnetic Schr\"{o}dinger operators with complex electric potentials. Under suitable conditions on the magnetic and electric potentials, we prove a resolvent estimate for spectral parameters in an unbounded parabolic…
A recent conjecture of Morita predicts a lower bound in temperature $T$ of a chaotic system, $T\geq (\hbar/2\pi)\Lambda$, $\Lambda$ being the Lyapunov exponent, which was demonstrated for a one dimensional inverse harmonic oscillator. In…