Quasilinear problems involving a perturbation with quadratic growth in the gradient and a noncoercive zeroth order term

In this paper we consider the problem u in H^1_0 (Omega), - div (A(x) Du) = H(x, u, Du) + f(x) + a_0 (x) u in D'(Omega), where Omega is an open bounded set of R^N, N \geq 3, A(x) is a coercive matrix with coefficients in L^\infty(Omega), H(x, s, xi) is a Carath\'eodory function which satisfies for some gamma > 0 -c_0 A(x) xi xi \leq H(x, s, xi) sign (s) \leq gamma A(x) xi xi a.e. x in Omega, forall s in R, forall xi in R^N, f belongs to L^{N/2} (Omega), and a_0 \geq 0 to L^q (Omega ), q > N/2. For f and a_0 sufficiently small, we prove the existence of at least one solution u of this problem which is moreover such that e^{delta_0 |u|} - 1 belongs to H^1_0 (Omega) for some delta_0 \geq gamma, and which satisfies an a priori estimate.Comment: 37 pages, 2 figure

Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u=0u=0 in a domain with many small holes

We perform the homogenization of the semilinear elliptic problem \begin{equation*} \begin{cases} u^\varepsilon \geq 0 & \mbox{in} \; \Omega^\varepsilon,\\ \displaystyle - div \,A(x) D u^\varepsilon = F(x,u^\varepsilon) & \mbox{in} \; \Omega^\varepsilon,\\ u^\varepsilon = 0 & \mbox{on} \; \partial \Omega^\varepsilon.\\ \end{cases} \end{equation*} In this problem $F(x,s)$ is a Carath\'eodory function such that $0 \leq F(x,s) \leq h(x)/\Gamma(s)$ a.e. $x\in\Omega$ for every $s > 0$, with $h$ in some $L^r(\Omega)$ and $\Gamma$ a $C^1([0, +\infty[)$ function such that $\Gamma(0) = 0$ and $\Gamma'(s) > 0$ for every $s > 0$. On the other hand the open sets $\Omega^\varepsilon$ are obtained by removing many small holes from a fixed open set $\Omega$ in such a way that a "strange term" $\mu u^0$ appears in the limit equation in the case where the function $F(x,s)$ depends only on $x$. We already treated this problem in the case of a "mild singularity", namely in the case where the function $F(x,s)$ satisfies $0 \leq F(x,s) \leq h(x) (\frac 1s + 1)$. In this case the solution $u^\varepsilon$ to the problem belongs to $H^1_0 (\Omega^\varepsilon)$ and its definition is a "natural" and rather usual one. In the general case where $F(x,s)$ exhibits a "strong singularity" at $u = 0$, which is the purpose of the present paper, the solution $u^\varepsilon$ to the problem only belongs to $H_{\tiny loc}^1(\Omega^\varepsilon)$ but in general does not belongs to $H^1_0 (\Omega^\varepsilon)$ any more, even if $u^\varepsilon$ vanishes on $\partial\Omega^\varepsilon$ in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results

A semilinear elliptic equation with a mild singularity at u=0u=0: existence and homogenization

In this paper we consider semilinear elliptic equations with singularities, whose prototype is the following \begin{equation*} \begin{cases} \displaystyle - div \,A(x) D u = f(x)g(u)+l(x)& \mbox{in} \; \Omega,\\ u = 0 & \mbox{on} \; \partial \Omega,\\ \end{cases} \end{equation*} where $\Omega$ is an open bounded set of $\mathbb{R}^N,\, N\geq 1$, $A\in L^\infty(\Omega)^{N\times N}$ is a coercive matrix, $g:[0,+\infty)\rightarrow [0,+\infty]$ is continuous, and $0\leq g(s)\leq {{1}\over{s^\gamma}}+1$ $\forall s>0$, with $0<\gamma\leq 1$ and $f,l \in L^r(\Omega)$, $r={{2N}\over{N+2}}$ if $N\geq 3$, $r>1$ if $N=2$, $r=1$ if $N=1$, $f(x), l(x)\geq 0$ a.e. $x \in \Omega$. We prove the existence of at least one nonnegative solution and a stability result; moreover uniqueness is also proved if $g(s)$ is nonincreasing or "almost nonincreasing". Finally, we study the homogenization of these equations posed in a sequence of domains $\Omega^\epsilon$ obtained by removing many small holes from a fixed domain $\Omega$

Semilinear problems with right-hand sides singular at u = 0 which change sign

The present paper is devoted to the study of the existence of a solution u for a quasilinear second order differential equation with homogeneous Dirichlet conditions, where the right-hand side tends to infinity at u = 0u=0. The problem has been considered by several authors since the 70's. Mainly, nonnegative right-hand sides were considered and thus only nonnegative solutions were possible. Here we consider the case where the right-hand side can change sign but is non negative (finite or infinite) at u = 0u=0, while no restriction on its growth at u = 0u=0 is assumed on its positive part. We show that there exists a nonnegative solution in a sense introduced in the paper; moreover, this solution is stable with respect to the right-hand side and is unique if the right-hand side is nonincreasing in u. We also show that if the right-hand side goes to infinity at zero faster than 1/ |u|1/∣u∣, then only nonnegative solutions are possible. We finally prove by means of the study of a one-dimensional example that nonnegative solutions and even many solutions which change sign can exist if the growth of the right-hand side is 1/ |u|\right.^{\gamma }\right. with 0 < \gamma < 10<γ<1

Semi-strong convergence of sequences satisfying variational inequality

International audienceIn this paper we study the properties of any sequence (un)n ≥ 1 weakly converging to a nonnegative function u in W1,p0(Ω), p>1, and satisfying a variational inequality of type -div(an(.,∇ un))≥ fn, where (an)n ≥ 1 is a suitable sequence of monotone operators and (fn)n ≥ 1 is any strongly convergent sequence in the dual space W-1,p'(Ω). We prove that the sequence (un - (1-ε)u)- strongly converges to 0 in W1,p0(Ω) for any ε>0. We show by a counter-example that the result does not hold true if ε=0. A remarkable corollary of these strong ε-convergences is that the sequence(u n)n ≥ 1 satisfies, up to a subsequence, a kind of semi-strong convergence: (un)n ≥ 1 can be bounded from below by a sequence which converges to the same limit u but strongly in W1,p0(Ω). We also give an example of a nonnegative weakly convergent sequence which does not satisfy this semi-strong convergence property and hence cannot satisfy any variational inequality of the previous type. Finally, in the linear case of a sequence of highly oscillating matrices, we improve the strong $\varepsilon$-convergences by replacing the arbitrary small constant ε>0 by a sequence(ε n)n ≥ 1 converging to 0

Comportement asymptotique d’une poutre élastique fixée sur une petite partie de l’une de ses extremités

We study the asymptotic behavior of the solution of an anisotropic, heterogeneous, linearized elasticity problem in a cylinder whose diameter ε tends to zero. The cylinder is assumed to be fixed (homogeneous Dirichlet boundary condition) on the whole of one of its extremities, but only on a small part (of size εrε) of the second one; the Neumann boundary condition is imposed on the remainder of the boundary. We show that the result depends on rε, and that there are 3 critical sizes, namely rε=ε3, rε=ε, and rε=ε1/3, and in total 7 different regimes. We also prove a corrector result for each behavior of rε.Nous étudions le comportement asymptotique de la solution d'un problème d'élasticité linéaire anisotrope et hétérogène dans un cylindre dont le diamètre ε tend vers zéro. Le cylindre est fixé (condition de Dirichlet homogène) sur la totalité de l'une de ses extrémités, mais seulement sur une petite partie (de taille εrε) de l'autre base ; sur le reste de la frontière on a la condition de Neumann. Nous montrons que le résultat depend de rε, et qu'il existe 3 tailles critiques, à savoir rε=ε3, rε=ε et rε=ε1/3, et au total 7 comportements différents. Nous donnons un résultat de correcteur pour tous les comportements de rε.Dirección General de Investigació

Comportamiento asintótico de una viga elástica fijada en pequeñas zonas de uno de sus extremos

Estudiamos el comportamiento asintótico de una viga elástica delgada cuando su anchura, ε, tiende a cero. La viga está fijada en la totalidad de una de sus bases, mientras que en la otra, sólo lo está en la unión de N pequeñas zonas de talla εrε , r ε tendiendo a cero. Sobre el resto de la frontera se impone una condición de Neumann. El comportamiento depende de r ε , el número de zonas de fijación y su distribución. Para N = 1 aparecen tres tallas críticas, ε 3 , ε y ε 1/3 y por tanto siete regímenes distintos. Si r ε ¿ ε 3 el comportamiento es el mismo que cuando no existe la pequeña zona de sujeción. Si r ε À ε 1/3 el comportamiento es el que obtendríamos si fijáramos en toda la base. En los demás casos aparecen comportamientos intermedios. Para N ≥ 2 el resultado es diferente. Así, si las zonas se concentran alrededor de tres puntos no alineados sólo aparecen dos tallas críticas, ε 3 y ε. Esto prueba que es preferible fijar la viga alrededor de tres puntos no alineados de una base a hacerlo alrededor de tan sólo uno, aún cuando usemos una zona de mucho mayor grosor

A model for two coupled turbulent fluids. Part II: numerical analysis of a spectral discretization

We consider a system of equations that models the stationary flow of two immiscible turbulentfluids on adjacentsubdomains. The equations are coupled by nonlinear boundary conditions on the interface which is here a fixed given surface. We propose a spectral discretization of this problem and perform its numerical analysis. The convergence of the method is proven in the two-dimensional case, together with optimal error estimates.Iberdrola Visiting Professors ProgrammeMinisterio de Ciencia y Tecnologí