arXiv subjects
Publications and source records attributed to Takanobu Hara.
We investigate the Cauchy-Dirichlet problem for linear parabolic equations in divergence form. Under mild assumptions on the source term and the domain, we prove the existence of globally Hölder continuous solutions. Notably, our results accommodate data exhibiting singularities nearly as critical as the inverse square of the distance from the boundary.
We prove the existence of globally Hölder continuous solutions to certain elliptic partial differential equations with lower-order terms. Our result is applicable to coefficients controlled by a negative power of the distance from the boundary of the domain and significantly improves Theorem 8.30 in Gilbarg and Trudinger (1983). The proof is derived by applying the strategy of Ancona (1986) to a new Morrey-type space.
We establish an existence result for globally continuous weak solutions to elliptic equations of the $p$-Poisson type. This result significantly improves Theorem 8.30 in Gilbarg-Trudinger (1983) and offers a novel contribution for the classical Poisson equation on Lipschitz domains, ensuring global Hölder continuity of solutions under a minimal assumption on the right-hand side. Applications of this result to embedding theorems are also discussed.
This is a progress report on study of uniformly elliptic Poisson-type equations on domains with capacity density conditions (CDC domains). We give a brief summary of known facts of CDC domains, including Hardy's inequality, and review a previous work of existence of globally Hölder continuous solutions. Additionally, we apply the result to homogenization problems of $ε$-periodic coefficients and present a convergence rate estimate of $L^{\infty}$ norms.
In potential theory, use of barriers is one of the most important techniques. We construct strong barriers for weighted quasilinear elliptic operators. There are two applications: (i) solvability of Poisson-type equations with boundary singular data, and (ii) a geometric version of Hardy inequality. Our construction method can be applied to a general class of divergence form elliptic operators on domains with rough boundary.
We discuss the solvability of Dirichlet problems of the type $- Δ_{p, w} u = σ$ in $Ω$; $u = 0$ on $\partial Ω$, where $Ω$ is a bounded domain in $\mathbb{R}^{n}$, $Δ_{p, w}$ is a weighted $(p, w)$-Laplacian and $σ$ is a nonnegative locally finite Radon measure on $Ω$. We do not assume the finiteness of $σ(Ω)$. We revisit this problem from a potential theoretic perspective and provide criteria for the existence of solutions by $L^{p}(w)$-$L^{q}(σ)$ trace inequalities or capacitary conditions. Additionally, we apply the method to the singular elliptic problem $- Δ_{p, w} u = σu^{- γ}$ in $Ω$; $u = 0$ on $\partial Ω$ and derive connection with the trace inequalities.
We consider model semilinear elliptic equations of the type \[ \begin{cases} - \mathrm{div} (A(x) \nabla u) = f u^{- λ}, \quad u > 0 \quad \text{in} \ Ω, \\ u \in H_{0}^{1}(Ω), \end{cases} \] where $Ω$ is a bounded domain in $\mathbf{R}^{N}$, $N \ge 1$, $A \in L^{\infty}(Ω)^{N \times N}$ is a coercive matrix, $0 < λ\le 1$ and $f$ is a nonnegative function in $L^{1}_{loc}(Ω)$, or more generally, nonnegative Radon measure on $Ω$. We discuss $H^{1}$-stability of $u$ under a minimal assumption on $f$. Additionally, we apply the result to homogenization problems.
We consider the existence of positive solutions to weighted quasilinear elliptic differential equations of the type \[ \begin{cases} - Δ_{p, w} u = σu^{q} & \text{in $Ω$}, \\ u = 0 & \text{on $\partial Ω$} \end{cases} \] in the sub-natural growth case $0 < q < p - 1$, where $Ω$ is a bounded domain in $\mathbb{R}^{n}$, $Δ_{p, w}$ is a weighted $p$-Laplacian, and $σ$ is a nonnegative (locally finite) Radon measure on $Ω$. We give criteria for the existence problem. For the proof, we investigate various properties of $p$-superharmonic functions, especially the solvability of Dirichlet problems with infinite measure data.
We study the existence of positive solutions to quasilinear elliptic equations of the type \[ -Δ_{p} u = σu^{q} + μ\quad \text{in} \ \mathbb{R}^{n}, \] in the sub-natural growth case $0 < q < p - 1$, where $Δ_{p}u = \nabla \cdot ( |\nabla u|^{p - 2} \nabla u )$ is the $p$-Laplacian with $1 < p < n$, and $σ$ and $μ$ are nonnegative Radon measures on $\mathbb{R}^{n}$. We construct minimal generalized solutions under certain generalized energy conditions on $σ$ and $μ$. To prove this, we give new estimates for interaction between measures. We also construct solutions to equations with several sub-natural growth terms using the same methods.