We consider the Dirichlet problem for a class of Hamilton-Jacobi equations and we show that if the viscosity solution is semiconcave near the boundary then it is globally semiconcave. As an application, we discuss the particular case of eikonal-type equations. Finally, we describe a sort of stability property for the singular set of the semiconcave viscosity solution of Hamilton-Jacobi equations.

This work is concerned with nonlinear stability properties of periodic travelling wave solutions of the Hirota-Satsuma system $$ \begin{cases} u_t+ u_{xxx}+6u_xu=2bvv_x\\ v_t+ v_{xxx}+ 3uv_x=0 \end{cases} $$ posed in $\mathbb R$ with $b>0$. We prove that this system is globally well posed in $L^2_{per}([0,L])\times H^1_{per}([0,L])$ by using Bourgain's space framework. Also shown is the existence of at least two nontrivial smooth curves of periodic travelling wave solutions depending on the classical Jacobian elliptic functions. We find dnoidal and cnoidal waves solutions. Then we prove the nonlinear stability of the dnoidal waves solutions in the energy space $L^2_{per}([0,L])\times H^1_{per}([0,L])$. The Floquet theory is used to obtain a detailed spectral analysis of the Jacobian form of Lam\'e's equation which is required in our stability theory.

We prove, by adapting the method of Colliander-Kenig [9], local well posedness of the initial-boundary-value problem for the one-dimensional nonlinear Schrödinger equation $i\partial_tu +\partial_x^2u +\lambda u|u|^{\alpha-1}=0$ on the half-line under low boundary regularity assumptions.

In this article, we consider the following problem $$(P)\hspace{3cm}\left\{ \begin{array}{cllll}\left. \begin{array}{rllll} -\Delta u & = & \mu u|u|^{p}e^{u^{2}} +\lambda h(x)\\ u & > & 0 \end{array}\right\} \;\; \text{in} \; \Omega, \\ u\;\;=\;\; 0 \;\;\text{on}\;\; \partial \Omega,\hspace{1.355cm} \end{array}\right.\hspace{2cm} $$ where $0\le p<\infty, \mu,\lambda >0, \Omega\subset \mathbb R^{2}$ is a bounded domain and $h \geq 0$ in $\Omega$ with $\| h \|_{L^{2}(\Omega)}=1.$ We show that there exist real numbers $0<\lambda_{*} \le \lambda^{*}$ such that the above problem admits at least two solutions for all $\lambda\in (0,\lambda_{*})$ and no solution for $\lambda > \lambda^{*}$.

Thanks to isomorphism results for Laplace's operator in weighted Sobolev spaces, we derive some applications. Various related spaces, intersection of Hardy spaces and weighted Sobolev spaces, are also considered. We show that the Riesz potentials spaces, consisting of all Riesz potentials $I_\alpha$ of order $\alpha$ of $L^p$ functions, with $1<p<\infty$, are equivalent to some appropriate weighted Sobolev spaces. It is known that if $ 1< p\le q<\infty$ and if $w$ and $v^{1-p'}$ satisfy the reverse doubling condition, then $I_\alpha$ satisfy the following continuity property: $$ \Big (\int_{\mathbb R }|I_\alpha f(x)|^q w(x) dx\Big )^{1/q}\le C \Big (\int_{\mathbb R }|f(x)|^p v(x) dx\Big )^{1/p}, \tag*{$(\star)$} $$ if and only if the weights satisfy the $(A_{p,q})$ condition $$ \frac {1}{|Q|^{1-\alpha/ {\rm N}}} \Big (\int_Q w dx\Big )^{1/q}\Big (\int_Q v^{1-p'} dx\Big ) ^{1/p'}\le C \ \ \ \hbox{ for all cubes } Q\subset\mathbb R . \tag*{$(A_{p,q})$} $$ A similar property holds for the Riesz transforms $R_j$ with $j = 1,\ldots , {\scriptstyle N}$: $$ \int_{\mathbb R }|R_j f(x)|^p u(x) dx\le C \int_{\mathbb R }|f(x)|^p u(x) dx, \tag*{$(\star\star)$} $$ if and only if the weight $u$ satisfies the Muckenhoupt $A_p$ condition: $$ (\dfrac{1}{|Q|}\int_Q u dx)^{1/p}(\dfrac{1}{|Q|}\int_Q u^{1-p'} dx) ^{1/p'}\le C \ \ \ \hbox{ for all cubes } Q\subset\mathbb R . \tag*{$(A_p)$} $$ Here, $f$ belongs to a weighted Sobolev space $W$. We show that even if these conditions are not satisfied, the continuity property of $I_\alpha$ and $R_j$ still holds in appropriate subspaces of $W$. By interpolation, a weighted version of the Sobolev imbedding theorem is also proved.