We give a self-contained and short proof for the existence, uniqueness and measurability of the $p$-harmonious functions in the case $2\le p < \infty$. The proofs only use elementary analytic tools. As a consequence, we obtain existence, uniqueness and measurability of value functions for the tug-of-war game with noise.

We prove that the streamlines and the profile of periodic gravity water waves traveling over a flat bed with wavespeed which exceeds the horizontal velocity of all fluid particles are real-analytic graphs if the vorticity function is merely integrable.

In this article, we establish in the radial framework the $H^1$-scattering for the critical 2-D nonlinear Schrödinger equation with exponential growth. Our strategy relies on both the a priori estimate derived in [10, 23] and the characterization of the lack of compactness of the Sobolev embedding of $H_{rad}^1(\mathbb R^2)$ into the critical Orlicz space ${\mathcal L}(\mathbb R^2)$ settled in [4]. The radial setting, and particularly the fact that we deal with bounded functions far away from the origin, occurs in a crucial way in our approach.

In this paper, we use probabilistic arguments (Tug-of-War games) to obtain the existence of viscosity solutions to a parabolic problem of the form $$ \begin{cases} K_{(x,t)}(D u )u_t (x,t)= \tfrac12 \langle D^2 u J_{(x,t)}(D u ),J_{(x,t)}(D u) (x,t)\rangle & \mbox{in } \Omega_T,\\ u(x,t)=F(x) & \mbox{on }\Gamma, \end{cases} $$ where $\Omega_T=\Omega\times(0,T]$ and $\Gamma$ is its parabolic boundary. This problem can be viewed as a version with spatial and time dependence of the evolution problem given by the infinity Laplacian, $$ u_t (x,t)= \langle D^2 u (x,t) \frac{D u}{|Du|} (x,t),\, \frac{D u}{|Du|} (x,t)\rangle\, . $$

We consider nonlocal elliptic boundary value problems of the form $$ -\mathcal{A}(x,u) \bigtriangleup u = \lambda f(u) $$ with Dirichlet boundary conditions. We show that if $f$ has $n$ loops, then the problem has at least $n$ distinct solutions, by using very simple comparison principles. Also, we study the asymptotic behavior of such solutions as the parameter $\lambda $ tends to $\infty$.

We consider the initial value problem for a three-component system of quadratic nonlinear Schrödinger equations with mass resonance in two space dimensions. Under a suitable condition on the coefficients of the nonlinearity, we will show that the solution decays strictly faster than ${O(t^{-1})}$ as $t \to +\infty$ in $L^{\infty}$ by providing an enhanced decay estimate of order $O((t\log t)^{-1})$. Differently from the previous works, our approach does not rely on the explicit form of the asymptotic profile of the solution at all.

In this paper, we consider the boundary value problem of Stokes operators with variable viscosity in the case of free boundary condition in a uniform $W^{2-1/r}_r$ domain of $N$-dimensional Euclidean space $\mathbb R^N$ ($N\geq 2$, $N < r < \infty$). We prove the $\mathcal R$-boundedness of solution operators with spectral parameter $\lambda$ varying in a sector $\Sigma_{\epsilon, \lambda_0} = \{\lambda \in \mathbf C : | \arg\lambda| \leq \pi-\epsilon, \,\, | \lambda| \geq \lambda_0\}$, from which we can deduce the $L_p$-$L_q$ maximal regularity as well as the generation of analytic semigroup for the time dependent problem. The problem of this type arises in the mathematical study of the incompressible viscous fluid flow with free surface. The essential assumption of this paper is the unique solvability of the weak Dirichlet-Neumann problem, namely, it is assumed the unique existence of solution $p \in \mathcal W^1_q(\Omega)$ to the variational problem: $(\nabla p, \nabla\varphi)_\Omega = (f, \nabla\varphi)_\Omega$ for any $\varphi \in \mathcal W^1_{q'}(\Omega)$ with $1 < q < \infty$ and $q' = q/(q-1)$, where $\mathcal W^1_q(\Omega)$ is a closed subspace of $\hat W^1_{q,\Gamma}(\Omega) = \{p \in L_{q, {\rm loc}} (\Omega) : \nabla p \in L_q(\Omega)^N, \,\, p|_\Gamma = 0\}$ with respect to gradient norm $\|\nabla\cdot\|_{L_q(\Omega)}$ that contains a space $W^1_{q,\Gamma}(\Omega) = \{p \in W^1_q(\Omega) : p|_\Gamma = 0\}$, and $\Gamma$ is one part of boundary on which free boundary condition is imposed. The unique solvability of such weak Dirichlet-Neumann problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to spectral parameter varying in $(\lambda_0, \infty)$, which was proved in Shibata [28]. Our assumption is satisfied for any $q \in (1, \infty)$ by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer with $\mathcal W^1_q(\Omega) = \hat W^1_{q, \Gamma}(\Omega)$, and by exterior domains with $\mathcal W^1_{q,\Gamma}(\Omega)$ $=$ the closure of $W^1_{q, \Gamma}(\Omega)$ with respect to the gradient norm. Combining the result in this paper with that in a forthcoming paper about the nonlinear problems, we can conclude that the unique existence of solutions to weak Dirichlet-Neumann problem implies a local in time unique existence theorem of strong solutions to the free boundary problem without surface tension taken into account for the Navier-Stokes equations in a uniform $W^{2-1/r}_r$ domain.

Recently, the first author studied in [17] the bifurcation of critical points of families of functionals on a Hilbert space, which are parametrized by a compact and orientable manifold having a non-vanishing first integral cohomology group. We improve this result in two directions: topologically and analytically. From the analytical point of view, we generalize it to a broader class of functionals. From the topological point of view, we allow the parameter space to be a metrizable Banach manifold. Our methods are, in particular, powerful if the parameter space is simply connected. As an application of our results, we consider families of geodesics in (semi-) Riemannian manifolds.

In this article, we present some results on the existence of positive and ground state solutions for the nonlinear Klein-Gordon-Maxwell equations. We introduce a general nonlinearity with subcritical and supercritical growth which does not require the usual Ambrosetti-Rabinowitz condition. The proof is based on variational methods and perturbation arguments.