We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated with fully nonlinear elliptic equations of order $2\,s$, with $s\in (1/2,\,1)$, and a coercive gradient term with subcritical power $0< p<2\,s$. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range $0< p<2\,s$, involving a continuum family of solutions and/or solutions blowing-up to $-\infty$ on the boundary. This is in striking difference with the local case studied by Lasry–Lions for the subquadratic case $1< p<2$.

In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle $\mathbb T,$ there is a function f holomorphic in the unit disc, having $\psi $ as radial limit a.e. on $\mathbb T.$ We consider an analogous problem for solutions f of homogenous elliptic equations $Pf=0$ and, in particular, for holomorphic functions on Riemann surfaces and harmonic functions on Riemannian manifolds.

We establish gradient estimates for solutions to the Dirichlet problem for the constant mean curvature equation in hyperbolic space. We obtain these estimates on bounded strictly convex domains by using the maximum principles theory of Φ-functions of Payne and Philippin. These estimates are then employed to solve the Dirichlet problem when the mean curvature H satisfies H < 1 under suitable boundary conditions.

In this paper, we analyse nonlocal equations in perforated domains. We consider nonlocal problems of the form $f(x) = \int \nolimits _{B} J(x-y) (u(y) - u(x)) {\rm d}y$ with x in a perforated domain $\Omega ^\epsilon \subset \Omega $. Here J is a nonsingular kernel. We think about $\Omega ^\epsilon $ as a fixed set Ω from where we have removed a subset that we call the holes. We deal both with the Neumann and Dirichlet conditions in the holes and assume a Dirichlet condition outside Ω. In the latter case we impose that u vanishes in the holes but integrate in the whole ℝN (B = ℝN) and in the former we just consider integrals in ℝN minus the holes ($B={\open R} ^N \setminus (\Omega \setminus \Omega ^\epsilon )$). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of $\Omega ^\epsilon $ has a weak limit, $\chi _{\epsilon } \rightharpoonup {\cal X}$ weakly* in L∞(Ω), we analyse the limit as ε → 0 of the solutions to the nonlocal problems proving that there is a nonlocal limit problem. In the case in which the holes are periodically removed balls, we obtain that the critical radius is of the order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behaviour of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems.

We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold M with only one end if M has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constant minimal graphic and p-harmonic functions on rotationally symmetric Cartan-Hadamard manifolds under optimal assumptions on the sectional curvatures.

In this paper, we study the fractional Dirichlet problem with the homogeneous exterior data posed on a bounded domain with Lipschitz continuous boundary. Under an extra assumption on the domain, slightly weaker than the exterior ball condition, we are able to prove existence and uniqueness of solutions which are Hölder continuous on the boundary. In proving this result, we use appropriate barrier functions obtained by an approximation procedure based on a suitable family of zero-th order problems. This procedure, in turn, allows us to obtain an approximation scheme for the Dirichlet problem through an equicontinuous family of solutions of the approximating zero-th order problems on ${\bar \Omega}$. Both results are extended to an ample class of fully non-linear operators.

In this paper we study a quasi-linear elliptic problem coupled with Dirichlet boundary conditions. We propose a new set of assumptions ensuring the existence of infinitely many solutions.

We show that the null limit hypothesis, in the definition of a barrier, can be relaxed for normal boundary points that satisfy a mild additional condition. We also give a simple necessary and sufficient condition for the regularity of semi-singular boundary points.

We model and study the genetic evolution and conservation of a population of diploid hermaphroditic organisms, evolving continuously in time and subject to resource competition. In the absence of mutations, the population follows a three-type, nonlinear birth-and-death process, in which birth rates are designed to integrate Mendelian reproduction. We are interested in the long-term genetic behavior of the population (adaptive dynamics), and in particular we compute the fixation probability of a slightly nonneutral allele in the absence of mutations, which involves finding the unique subpolynomial solution of a nonlinear three-dimensional recurrence relationship. This equation is simplified to a one-dimensional relationship which is proved to admit exactly one bounded solution. Adding rare mutations and rescaling time, we study the successive mutation fixations in the population, which are given by the jumps of a limiting Markov process on the genotypes space. At this time scale, we prove that the fixation rate of deleterious mutations increases with the number of already fixed mutations, which creates a vicious circle called the extinction vortex.

We prove the existence and uniqueness of harmonic maps between rotationally symmetric manifolds that are asymptotically hyperbolic.

We give a systematic treatment of caloric measure for arbitrary open sets. The caloric measure is defined only on the essential boundary of the set. Our main result gives criteria for the resolutivity of essential boundary functions, and their integral representation in terms of caloric measure. We also characterize the caloric measure null sets in terms of the boundary singularities of nonnegative supertemperatures.

As in preceding papers inwhich we studied the limits of penalized 1-dimensional Wienermeasures with certain functionals Γt, we obtain here theexistence of the limit, as t → ∞, of d-dimensional Wienermeasures penalized by a function of the maximum up to time t ofthe Brownian winding process (for d = 2), or in {d}≥ 2dimensions for Brownian motionprevented to exit a cone before time t. Various extensions of these multidimensional penalisations arestudied, and the limit laws are described. Throughout this paper, the skew-product decomposition ofd-dimensional Brownian motion plays an important role.

The Dirichlet problem for the Laplace equation in a connected-plane region with cuts is studied. The existence of a classical solution is proved by potential theory. The problem is reduced to a Fredholm equation of the second kind, which is uniquely solvable.

Given an open set $\Omega$ of compact closure in $\mathbb{R}^m$, the classical Dirichlet problem is to extend a given continuous function $\psi : \partial \Omega \to \mathbb{R}$ to a continuous function $\phi : \overline \Omega \to \mathbb{R}$ such that $\phi$ is harmonic (that is, satisfies the Laplace equation) in $\Omega$. The set $\Omega$ is termed regular if the Dirichlet problem has a (necessarily unique) solution for any continuous boundary function $\psi$. For example, every simply connected planar domain is regular (but may have a 'bad' boundary $\partial\Omega$, for instance, a fractal).

In this article it is shown that, when $\Omega$ is regular (in the above sense), every continuous map $\psi$ from $\partial\Omega$ to a simply connected complete Riemannian manifold $(N, h)$ of sectional curvature at most 0 has a unique continuous extension $\phi : \overline\Omega \to N$ which is harmonic in $\Omega$. This is done with $\mathbb{R}^m$ replaced more generally by an $m$-dimensional Riemannian manifold $(M, g)$.

The proof relies on the unique solvability of the corresponding variational Dirichlet problem (for any open set $\Omega \Subset M$). And for that, the above target manifold $N$ can be replaced more generally by any simply connected complete geodesic space $Y$ of curvature at most 0 in the sense of A. D. Alexandrov. Assuming that $M$ satisfies the Poincaré inequality, we show that, for any map $\psi : M \to Y$ of finite energy in the sense of N. J. Korevaar and R. M. Schoen, there exists a unique map $\phi : M \to Y$ with $\phi = \psi$ on $M \setminus \Omega$ such that $\phi$ minimizes the energy of all maps $M \to Y$ which agree with $\psi$ on $M \setminus \Omega$. If $\Omega$ is regular then $\phi$ is continuous at any point of $\partial\Omega$ at which $\psi$ is continuous. For a Lipschitz (and hence regular) domain $\Omega \Subset M$, existence and uniqueness of the variational solution $\phi$ was obtained by Korevaar and Schoen, and earlier for suitable polyhedral targets $Y$ by Gromov and Schoen.

Instead of the Riemannian manifold $M$ our domain space can still more generally be an admissible Riemannian polyhedron (as studied in the recent Cambridge Tract by J. Eells and the present author); the variational solution $\phi$ is then in general only Hölder continuous in $\Omega$.

The proofs of the stated results of this article rely in part on potential theory relative to the fine topology of H. Cartan.

The singular boundary-value problem $(g(x'(t)))'=\mu f(t,x(t),x'(t))$, $x(0)=x(T)=0$ and $\max\{x(t):0\le t\le T\}=A$ is considered. Here $\mu$ is the parameter and the negative function $f(t,u,v)$ satisfying local Carathéodory conditions on $[0,T]\times(0,\infty)\times(\mathbb{R}\setminus\{0\})$ may be singular at the values $u=0$ and $v=0$ of the phase variables $u$ and $v$. The paper presents conditions which guarantee that for any $A>0$ there exists $\mu_A>0$ such that the above problem with $\mu=\mu_A$ has a positive solution on $(0,T)$. The proofs are based on the regularization and sequential techniques and use the Leray–Schauder degree and Vitali’s convergence theorem.

AMS 2000 Mathematics subject classification: Primary 34B16; 34B18

In this paper, we shall study the Dirichlet problem for the minimal surfaces equation. We prove some results about the boundary behaviour of a solution of this problem. We describe the behaviour of a non-converging sequence of solutions in term of lines of divergence in the domain. Using this second result, we build some solutions of the Dirichlet problem on unbounded domain. We then give a new proof of the result of Cosín and Ros concerning the Plateau problem at infinity for horizontal ends.

AMS 2000 Mathematics subject classification: Primary 53A10

Given a homogeneous elliptic partial differential operator $L$ with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain $\Omega$ in ${{\mathbf{R}}^{n}}$ and which belong locally to a Banach space $V$, we consider the problem of approximating in the norm of $V$ the functions in this class by “analytic” and “meromorphic” solutions of the equation $Lu\,=\,0$. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces $V$ and operators $L$. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

We study a stochastic differential equation model of prey-predator evolution. To keep in line with a known deterministic model we include the social and interaction terms with the drift in our model, and the randomness arises as fluctuations in the ecosystem. The notion of equilibrium population level and special types of fluctuations force us to work with degenerate elliptic operators. We consider the propagation of the population system in a sufficiently large but bounded domain. This enables us to look at not only the population extinction, but also the explosion beyond a certain level. Both extinction and explosion are possible; and, when they are not, we show that the population asymptotically reaches the equilibrium level. We show that the extinction, explosion and saturation probabilities satisfy, as functions of the initial population size, an integral equation arising out of a Dirichlet problem for a non-degenerate elliptic equation; and these probabilities are also smooth solutions of the Dirichlet problems. They are also used to express the solution of another Dirichlet problem for a degenerate elliptic equation.