Regularity in a one-phase free boundary problem for the fractional Laplacian

For a one-phase free boundary problem involving a fractional Laplacian, we prove that “flat free boundaries” are C^1,α$C^{1,\alpha }$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.     

Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.     

Regularity for the fractional Gelfand problem up to dimension 7

We study the problem (−Δ)^su=λe^u${(-\mathrm{\Delta })}^{s}u=\lambda e^u$ in a bounded domain Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$, where λ   is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions n≤7$n\le 7$ for all s∈(0,1)$s\in (0,1)$ whenever Ω   is, for every i=1,...,n$i=1,...,n$, convex in the _xi$x_i$-direction and symmetric with respect to {_xi=0}$\{x_i=0\}$. The same holds if n=8$n=8$ and s?0.28206...$s?0.28206...$, or if n=9$n=9$ and s?0.63237...$s?0.63237...$. These results are new even in the unit ball Ω=_B1$\Omega =B_1$.     

Regularity in Capacity and the Dirichlet Laplacian

Given an open set in , we prove that every function in is zero everywhere on the boundary if and only if is regular in capacity. If in addition is bounded, then it is regular in capacity if and only if the mapping from into is injective, where denotes the Perron solution of the Dirichlet problem. Let be the set of all open subsets of which are regular in capacity. Then one can define metrics and on only involving the resolvent of the Dirichlet Laplacian. Convergence in those metrics will be defined to be the local/global uniform convergence of the resolvent of the Dirichlet Laplacian applied to the constant function . We prove that the spaces and are complete and contain the set of all open sets which are regular in the sense of Wiener (or Dirichlet regular) as a closed subset.     

Regularity of solutions to Dirichlet boundary value problem in domains on a manifold

The regularity of solutions to the Dirichlet boundary value problem is studied for elliptic differential operators of order 2m defined on subdomains of a manifold. Relationships between the smoothness of the right hand side, the boundary, and the solutions are obtained.     

An extension problem for the CR fractional Laplacian

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre. We also prove an energy identity that yields a sharp trace Sobolev embedding.     

Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian

The paper is concerned with the Dirichlet problem

The Dirichlet problem for the Kohn Laplacian on the Heisenberg group,II

Let L = ∑_{j = 1}^mX_j² be sum of squares of vector fields in $\text{R}$ⁿ satisfying a Hörmander condition of order 2: span{X_j, [X_i, X_j]} is the full tangent space at each point. A point x??D of a smooth domain D is characteristic if X₁,…, X_m are all tangent to ?D at x. We prove sharp estimates in non-isotropic Lipschitz classes for the Dirichlet problem near (generic) isolated characteristic points in two special cases: (a) The Grushin operator $\text{?}{}^2\text{?x}{}^2\text{+ x}{}^2\text{?}{}^2\text{?t}{}^2$ in $\text{R}$². (b) The real part of the Kohn Laplacian on the Heisenberg group $\text{∑}{}_{\mathrm{j\; ?\; 1}}{}^{\mathrm{n}}\text{(}\text{?}\text{?x}{}_{\mathrm{j}}\text{+ 2y}{}_{\mathrm{j}}\text{?}\text{?t}\text{)}{}^2\text{+ (}\text{?}\text{?y}{}_{\mathrm{j}}\text{? 2x}{}_{\mathrm{j}}\text{?}\text{?t}\text{)}{}^2$ in $\text{R}$^{2n + 1}. In contrast to non-characteristic points, C^∞ regularity may fail at a characteristic point. The precise order of regularity depends on the shape of ?D at x.     

The Dirichlet problem for the Kohn Laplacian on the Heisenberg group,I

For (x,y,t)∈$\text{R}$ⁿ × $\text{R}$ⁿ × $\text{R}$, denote $\text{X}{}_{\mathrm{j}}\text{=}\text{?}\text{?x}{}_{\mathrm{j}}\text{+ 2y}{}_{\mathrm{j}}\text{?}\text{?t}\text{, y}{}_{\mathrm{j}}\text{=}\text{?}\text{?y}{}_{\mathrm{j}}\text{? 2x}{}_{\mathrm{j}}\text{?}\text{?t}$ and $\text{L}{}_{\mathrm{\alpha }}\text{=?}\text{1}\text{4}\text{∑}\text{j=1}\text{n}\text{X}{}_{\mathrm{j}}{}^2\text{+ Y}{}_{\mathrm{j}}{}^2\text{+}\text{?}\text{?t}$. When α = n ? 2q, $\text{L}$_a represents the action of the Kohn Laplacian □_b on q-forms on the Heisenberg group. For ?n < α < n, we construct a parametrix for the Dirichlet problem in smooth domains D near non-characteristic points of ?D. A point w of ?D is non-characteristic if one of X₁,…, X_n, Y₁,…, Y_n is transverse to ?D at w. This yields sharp local estimates in the Dirichlet problem in the appropriate non-isotropic Lipschitz classes. The main new tool is a “convolution calculus” of pseudo-differential operators that can be applied to the relevant layer potentials, for which the usual asymptotic composition formula is false. Characteristic points are treated in Part II.     

The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation

We study the existence, uniqueness and continuous dependence on initial data of the solution for a nonlocal Cahn-Hilliard equation with Dirichlet boundary condition on a bounded domain. Under a nondegeneracy assumption the solutions are classical but when this is relaxed, the equation is satisfied in a weak sense. Also we prove that there exists a global attractor in some metric space.     

Regularity up to the boundary for generalized solutions to the first boundary-value problem for doubly degenerate quasilinear parabolic equations

For quasilinear doubly degenerate parabolic equations it has been possible to establish the existence of nonnegative generalized solutions to the first boundary-value problem that are Hölder continuous up to the boundary of the cylinder Q_T=Ωx(0, T]. A typical example of an admissible equation is the equation of Newtonian polytropic filtration $$\frac{{\partial u}}{{\partial t}} - \frac{\partial }{{\partial x_i }}\left\{ {a_0 \left| u \right|^l \left| {u_x } \right|^{m - 2} u_{x_i } } \right\} = 0, a_0 > 0,l > 0,m > 2.$$ .     

Ramification of a multiple eigenvalue of the Dirichlet problem for the Laplacian under singular perturbation of the boundary condition

Translated from Matematicheskie Zametki, Vol. 52, No. 4, pp. 42–55, October, 1992.     

The extremal solution for the fractional Laplacian

We study the extremal solution for the problem \((-\Delta )^s u=\lambda f(u)\) in \(\Omega \) , \(u\equiv 0\) in \(\mathbb R ^n\setminus \Omega \) , where \(\lambda >0\) is a parameter and \(s\in (0,1)\) . We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions \(n<4s\) . We also show that, for exponential and power-like nonlinearities, the extremal solution is bounded whenever \(n<10s\) . In the limit \(s\uparrow 1\) , \(n<10\) is optimal. In addition, we show that the extremal solution is \(H^s(\mathbb R ^n)\) in any dimension whenever the domain is convex. To obtain some of these results we need \(L^q\) estimates for solutions to the linear Dirichlet problem for the fractional Laplacian with \(L^p\) data. We prove optimal \(L^q\) and \(C^\beta \) estimates, depending on the value of \(p\) . These estimates follow from classical embedding results for the Riesz potential in \(\mathbb R ^n\) . Finally, to prove the \(H^s\) regularity of the extremal solution we need an \(L^\infty \) estimate near the boundary of convex domains, which we obtain via the moving planes method. For it, we use a maximum principle in small domains for integro-differential operators with decreasing kernels.     

The regulator problem for parabolic equations with Dirichlet boundary control

This paper considers the regulator problem for a parabolic equation (generally unstable), defined on an open, bounded domain , with control functionu acting in the Dirichlet boundary condition: minimize the quadratic functional which penalizes theL ₂(0, ; L₂())-norm of the solutiony and theL ₂(0, ; L₂())-norm of the Dirichlet controlu. The paper is divided in two parts. Part I derives, in a constructive way, the algebraic Riccati equation satisfied by the candidate Riccati operator solution (unique in our case) and, moreover, studies the regularity properties of the optimal pairu ⁰, y⁰. Part II studies a Galerkin approximation of the regulator problem. It shows first the uniform analyticity and the uniform exponential stability of the underlying discrete (approximating) semigroups. Then it establishes the desired convergence properties, in particular, pointwise Riccati operators convergence and, as a final goal, convergence of the original dynamics acted upon by the discrete feedbacks.Research partially supported by the National Science Foundation under Grant DMS-8301668.     

Uniqueness for the homogeneous Dirichlet Willmore boundary value problem

We give a sufficient condition for curves on a plane or on a sphere such that if these give the boundary of a Willmore surface touching tangentially along the boundary the plane or the sphere respectively, the surface is necessarily a piece of the plane or a piece of the sphere. The condition we require is that the curves bound a strictly star-shaped domain with respect to the Euclidean geometry in the plane and with respect to the spherical geometry in the sphere, respectively.