Optimal control of semilinear elliptic equations in measure spaces

Optimal control problems in measure spaces governed by semilinear elliptic equations are considered. First order optimality conditions are derived and structural properties of their solutions, in particular sparsity, are discussed. Necessary and sufficient second order optimality conditions are obtained as well. On the basis of the sufficient conditions, stability of the solutions is analyzed. Highly nonlinear terms can be incorporated by utilizing an L∞(Ω) regularity result for solutions of the first order necessary optimality conditions.This author’s research was supported by Spanish Ministerio de Economía y Competitividad under project MTM2011-22711

Orientability and energy minimization in liquid crystal models

Uniaxial nematic liquid crystals are modelled in the Oseen-Frank theory through a unit vector field $n$. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which $n$ should be equivalent to -$n$. This symmetry is preserved in the constrained Landau-de Gennes theory that works with the tensor $Q=s\big(n\otimes n- \frac{1}{3} Id\big)$.We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural energy class $W^{1,2}$ the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains we completely characterise the instances in which the predictions of the constrained Landau-de Gennes theory differ from those of the Oseen-Frank theory

Supercritical biharmonic equations with power-type nonlinearity

The biharmonic supercritical equation $\Delta^2u=|u|^{p-1}u$, where $n>4$ and $p>(n+4)/(n-4)$, is studied in the whole space $\mathbb{R}^n$ as well as in a modified form with $\lambda(1+u)^p$ as right-hand-side with an additional eigenvalue parameter $\lambda>0$ in the unit ball, in the latter case together with Dirichlet boundary conditions. As for entire regular radial solutions we prove oscillatory behaviour around the explicitly known radial {\it singular} solution, provided $p\in((n+4)/(n-4),p_c)$, where $p_c\in ((n+4)/(n-4),\infty]$ is a further critical exponent, which was introduced in a recent work by Gazzola and the second author. The third author proved already that these oscillations do not occur in the complementing case, where $p\ge p_c$. Concerning the Dirichlet problem we prove existence of at least one singular solution with corresponding eigenvalue parameter. Moreover, for the extremal solution in the bifurcation diagram for this nonlinear biharmonic eigenvalue problem, we prove smoothness as long as $p\in((n+4)/(n-4),p_c)$

Lower bound for energies of harmonic tangent unit-vector fields on convex polyhedra

We derive a lower bound for energies of harmonic maps of convex polyhedra in $\R^3$ to the unit sphere $S^2,$ with tangent boundary conditions on the faces. We also establish that $C^\infty$ maps, satisfying tangent boundary conditions, are dense with respect to the Sobolev norm, in the space of continuous tangent maps of finite energy.Comment: Acknowledgment added, typos removed, minor correction

Dynamic Transitions for Quasilinear Systems and Cahn-Hilliard equation with Onsager mobility

The main objectives of this article are two-fold. First, we study the effect of the nonlinear Onsager mobility on the phase transition and on the well-posedness of the Cahn-Hilliard equation modeling a binary system. It is shown in particular that the dynamic transition is essentially independent of the nonlinearity of the Onsager mobility. However, the nonlinearity of the mobility does cause substantial technical difficulty for the well-posedness and for carrying out the dynamic transition analysis. For this reason, as a second objective, we introduce a systematic approach to deal with phase transition problems modeled by quasilinear partial differential equation, following the ideas of the dynamic transition theory developed recently by Ma and Wang

Local existence of analytical solutions to an incompressible Lagrangian stochastic model in a periodic domain

We consider an incompressible kinetic Fokker Planck equation in the flat torus, which is a simplified version of the Lagrangian stochastic models for turbulent flows introduced by S.B. Pope in the context of computational fluid dynamics. The main difficulties in its treatment arise from a pressure type force that couples the Fokker Planck equation with a Poisson equation which strongly depends on the second order moments of the fluid velocity. In this paper we prove short time existence of analytic solutions in the one-dimensional case, for which we are able to use techniques and functional norms that have been recently introduced in the study of a related singular model.Comment: 32 page

Controllability of the heat equation with an inverse-square potential localized on the boundary

This article is devoted to analyze control properties for the heat equation with singular potential $-\mu/|x|^2$ arising at the boundary of a smooth domain \Omega\subset \rr^N, $N\geq 1$. This problem was firstly studied by Vancostenoble and Zuazua [20] and then generalized by Ervedoza [10]in the context of interior singularity. Roughly speaking, these results showed that for any value of parameters $\mu\leq \mu(N):=(N-2)^2/4$, the corresponding parabolic system can be controlled to zero with the control distributed in any open subset of the domain. The critical value $\mu(N)$ stands for the best constant in the Hardy inequality with interior singularity. When considering the case of boundary singularity a better critical Hardy constant is obtained, namely $\mu_{N}:=N^2/4$. In this article we extend the previous results in [18],[8], to the case of boundary singularity. More precisely, we show that for any $\mu \leq \mu_N$, we can lead the system to zero state using a distributed control in any open subset. We emphasize that our results cannot be obtained straightforwardly from the previous works [20], [10].Comment: 32 page