We investigate the Dirichlet problem for the variational integral $J[u] = \int_{\Omega} f(\nabla u) \, dx$ with density $f$ of linear growth satisfying appropriate ellipticity conditions. We show that the relaxed problem admits a unique solution $u$ in the space of functions of bounded variation, if the set $\Gamma_0$ of convex points $x \in \partial\Omega$ is sufficiently large. For example, the inequality $\mathcal{H}^{n-1}(\Gamma_0)>\frac{2}{3}\mathcal{H}^{n-1}(\partial\Omega)$ is sufficient. Moreover, the minimizer $u$ is smooth in the interior of $\Omega$ and attains the prescribed boundary data at least on $\Gamma_0$ in the classical sense.
In this paper, we study the uniqueness of positive solutions to the semilinear elliptic Robin problem $$ \begin{cases} -\Delta u = u^p,&\text{in } \Omega,\\ u>0,&\text{in } \Omega,\\ \frac{\partial u}{\partial \nu} + \beta u = 0,&\text{on } \partial \Omega, \end{cases} $$ where $\beta>0$, $p$ is subcritical, and $\Omega$ is a bounded smooth domain. It is known that the uniqueness of the solution depends on the shape of the domain. Even if $\Omega$ is a ball, the problem is open for arbitrary $\beta>0$, since the method of moving planes does not work for Robin boundary conditions . By scaling arguments and a careful analysis of the linearized problem, we prove uniqueness for any $\beta>0$ provided that $p$ and $\Omega$ satisfy suitable conditions. Finally, we study the effects of concave and convex nonlinearities.
Let $\Omega\subset\mathbb R^n$ be a bounded domain, and let $f$ be a nonnegative, nondecreasing function satisfying the Keller-Osserman condition. We study boundary blow-up solutions of $\Delta u=f(u)$ in $\Omega$. Although existence is classical, uniqueness under these assumptions is known in balls but remains open even for smooth convex domains. We identify the normalized gradient $Q_u=|\nabla u|^2/(2F(u))$, $F'=f$, as a quantity governing uniqueness. Under a structural condition on $f$, a boundary blow-up solution $u$ is unique if $\limsup_{x\to\partial\Omega}Q_u(x)\le 1$, without any regularity assumption on $\partial \Omega$. For $C^{1,1}$ domains, assuming a growth condition on $f$, we prove $Q_u(x)\to 1$ for every boundary blow-up solution and hence obtain uniqueness under the structural condition. For convex domains, we prove $Q_u\le 1$ for the minimal boundary blow-up solution and obtain uniqueness when $\sqrt F$ is eventually convex, without imposing any additional boundary regularity.
We study global regularity of solutions to Dirichlet or Neumann elliptic problems in spherical sectors $S_{D,R}$ of radius $R>0$ in $\mathbb{R}^N, N\ge 2$, where $D$ is the bounded domain on the unit sphere $\mathbb{S}^{N-1}$ which spans the spherical sector. One of the main results shows that boundedness of the gradient of the solutions of Poisson equations holds whenever $\lambda_1(D)\ge N-1$, where $\lambda_1(D)$ is the first nontrivial eigenvalue of the Laplace Beltrami operator $-\Delta_{\mathbb{S}^{N-1}}$ on the domain $D$ with Dirichlet or Neumann boundary conditions on $\partial D$. As an example of Maz'ya shows, the condition on the eigenvalue is sharp. For general spherical sectors and for $p$-Laplacian equations, $p>1$ we prove weighted global lipschitzianity of the solutions, as well as second order regularity.
Carlo Alberto Antonini, F. Pacella, Camilla Chiara Polvara et al.· 0 citations
We identify a common convexity structure for three exponential Dirichlet problems on smooth uniformly strictly convex domains: the Liouville equation $\Delta u=e^u$, the real equation $\sigma_2(D^2u)=e^{2u}$, and its complex counterpart $\sigma_2(u_{i\bar j})=e^{2u}$. In each case $u<0$ in the domain and $u=0$ on the boundary. We prove that \[ w=-\operatorname{arcosh}(e^{-u/2}) \] is strictly convex in the underlying real variables. The argument combines domain deformation, constant-rank theory, inverse-convexity estimates, radial ball models, boundary strict convexity, and local $C^2$ stability.
In this work, we conduct a comprehensive study of problem \begin{equation*} \begin{cases} -\Delta_1 u + g(u)|Du| = h(u)f&\text{in }\Omega, u=0&\text{on } \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^N$ is a bounded Lipschitz domain, $f\in L^1(\Omega)$ is a nonnegative datum, and $g,h$ are nonnegative continuous functions on $(0,\infty)$ that may be singular at the origin. Under the minimal assumptions that $g$ is integrable near zero and $h$ is bounded at infinity, we explore the existence of a global $BV(\Omega)$ solution. Furthermore, a comparison principle is proved under suitable monotonicity assumptions on $h$. This framework avoids any growth restrictions on $h$ near the origin, thus allowing for highly singular terms. To handle these nonlinearities, we introduce a novel approach that takes advantage of the rigid structure of the 1-Laplacian operator.
This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta)^s u+ g|u|^{p-1}u= \lambda \frac{u}{|x|^{2s}}+f(x),$$ in a bounded domain $\Omega$ of $\mathbb{R}^N\,(N>2s)$, subject to the zero Dirichlet condition in $\mathbb{R}^N\setminus \Omega$, where $01$ and $f\in L^{(p+1)/p}_g(\Omega)$. Under certain integrability condition on $g$, the existence of solution is proven for every $\lambda\in \mathbb{R}$. Moreover, the regularity of solution is also obtained.
Rubén Fiñana, A. Molino· Nonlinear Analysis· 0 citations
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