Related papers: The two-phase problem for harmonic measure in VMO

We derive the long-time asymptotics for the Toda shock problem using the nonlinear steepest descent analysis for oscillatory Riemann--Hilbert factorization problems. We show that the half plane of space/time variables splits into five main…

We consider the balayage of a measure $\mu$ defined on a domain $\Omega$ onto its boundary $\partial \Omega$. Assuming that $\Omega$ has a corner of opening $\pi \alpha$ at a point $z_0 \in \partial \Omega$ for some $0 < \alpha \leq 2$ and…

We consider a two-dimensional equilibrium measure problem under the presence of quadratic potentials with a point charge and derive the explicit shape of the associated droplets. This particularly shows that the topology of the droplets…

We give bilateral pointwise estimates for positive solutions of the equation \begin{equation*} \left\{ \begin{aligned} -\triangle u & = \omega u \, \,& & \mbox{in} \, \, \Omega, \quad u \ge 0, \\ u & = f \, \, & &\mbox{on} \, \, \partial…

The anharmonic electron-phonon problem is solved in the infinite-dimensional limit using quantum Monte Carlo simulation. Charge-density-wave order is seen to remain at half filling even though the anharmonicity removes the particle-hole…

In this paper, we deal with the following double phase problem $$ \left\{\begin{array}{ll} -\mbox{div}\left(|\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right)=…

We consider the nonlinear and nonlocal problem $$ A_{1/2}u=|u|^{2^\sharp-2}u\ \text{in \Omega, \quad u=0 \text{on} \partial\Omega $$where $A_{1/2}$ represents the square root of the Laplacian in a bounded domain with zero Dirichlet boundary…

Let $u$ be a harmonic function in a $C^1$ domain $D\subset \mathbb{R}^d$, which vanishes on an open subset of the boundary. In this note we study its critical set $\{x \in \overline{D}: \nabla u(x) = 0 \}$. When $D$ is a $C^{1,\alpha}$…

We derive the quantitative modulus of continuity $$ \omega(r)=\left[ p+\ln \left( \frac{r_0}{r} \right) \right]^{-\alpha (n,p)}, $$ which we conjecture to be optimal, for solutions of the $p$-degenerate two-phase Stefan problem. Even in the…

We study non-supersymmetric truncations of $\omega$-deformed ${\cal N}=8$ gauged supergravity that retain a $U(1)$ gauge field and three scalars, of which two are neutral and one charged. We construct dyonic domain-wall and black hole…

We consider the problem $-\Delta u+\lambda u=u^{p-1}$, where $u\in H^1_0(\Omega)$ verifies $\|u\|_{L^2}=m>0$, and $\lambda\in [0,+\infty)$. Here, $\mathbb{R}^N\setminus\Omega$ is nonempty and compact. We prove the existence of a solution…

We investigate the effect of anharmonicity on the one-dimensional half-filled Holstein model by using the determinant quantum Monte Carlo method. By calculating the order parameters we find that with and without anharmonicity there is…

Our starting point is the measure $\epsilon_x-\alpha_x\rho_x^{\omega_1}+\beta_x\rho_x^{\omega_2}$, where $\rho_x^{\omega_i}$ is the harmonic measure relative to $x \in \omega_1 \subset \overline{\omega}_1 \subset \omega_2$ and $\omega_i$…

Let $N=(\Omega,\sigma)$ and $M=(\Omega^*,\rho)$ be doubly connected Riemann surfaces and assume that $\rho$ is a smooth metric with bounded Gauss curvature $\mathcal{K}$ and finite area. The paper establishes the existence of homeomorphisms…

In this paper, we consider the existence of nontrivial solutions to the following critical biharmonic problem with a logarithmic term \begin{equation*} \begin{cases} \Delta^2 u=\mu \Delta u+\lambda u+|u|^{2^{**}-2}u+\tau u\log u^2, \ \…

Let $\Omega\subset\mathbb R^{n+1}$ be an open set with $n$-AD-regular boundary. In this paper we prove that if the harmonic measure for $\Omega$ satisfies the so-called weak-$A_\infty$ condition, then $\Omega$ satisfies a suitable…

In this paper, we will study the partial regularity theorem for stationary harmonic maps from a Riemannian manifold into a Lorentzian manifold. For a weakly stationary harmonic map $(u,v)$ from a smooth bounded open domain…

We consider the equation $d^2\Delta u - u+ u^{\frac{n-k+2}{n-k-2}} =0\,\hbox{in}\Omega $, under zero Neumann boundary conditions, where $\Omega$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a…

In this article, we consider eigenfunctions $u$ of the bi-harmonic operator, i.e., $\triangle^2u=\lambda^2u$ on $\Omega$ with some homogeneous linear boundary conditions. We assume that $\Omega\subseteq\mathbb{R}^n$ ($n\geq2$) is a…

In a recent paper we have considered the long time asymptotics of the periodic Toda lattice under a short range perturbation and we have proved that the perturbed lattice asymptotically approaches a modulated lattice. In the present paper…