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This question is related to a prior question i asked, see Fourier Transform ; half space elliptic baby problem. Essentially I am asking the same question now but taking a lot more care. So lets examin …

I am trying to prove some monotonicity of a solution of a given pde; after considering a quantity like $ \phi(x) = x \cdot \nabla v(x)$ ($v$ is the solution of a given pde) I arrive at something alo …

Recall given any function $v(x)$ defined on $B$ (the unit ball centred at the origin in $ R^N$) we can write $$v(x) = \sum_{k=0}^\infty a_k(r) \psi_k(\theta)$$ where $ r=|x|$ and $ \theta = \frac …

I am interested when one can say the eigenvalues of an elliptic operator are real analytic with respect to a parameter. In particular I have seen many people say the first eigenvalue is analytic but …

Consider $ \Omega$ a smooth bounded domain in $ \mathbb R^N$. I am interested in the gap between the first and second eigenvalues of the operator $ -\Delta + V(x)$. Let $ \phi_1>0$ and $ \phi_2$ be …

I asked this question Poisson equation estimates near boundary a few days ago but haven't gotten any response. So I will ask a related question. Suppose $-\Delta u(x)=f(x)$ in $B_1^+$ in the (upp …

Fix $ \Omega$ a bounded smooth domain in $\mathbb R^N$ (take $N$ big) and let $ \frac{N+1}{2}<p<N$. We now consider nonnegative smooth functions $f$ such that $-\Delta u(x)=f(x) $ in $ \Omega$ with …

Let $B$ denote the unit ball centered at the origin in $R^N$ and take $N \ge 3$. Let $( \phi_k(\theta), \lambda_k)$ for $ k \ge 0$ denote the Eigenpairs of $ -\Delta_\theta$ on $S^{N-1}$ which are $L …

Suppose $N \ge 3$ and let $\Phi(x):= C_N |x|^{2-N}$ is the fundamental solution. Let $\Omega$ denote a bounded domain in $ R^N$. Consider $ -\Delta u(x) = f(x) $ in $\Omega$ with $u=0$ on $ \pa …

I am sure this is extremely well known but I have been digging a bit and I can't find what I need. Consider $B$ to be the unit ball in $ \mathbb R^N$ and consider the eigenvalue problem \begin{cases …

I asked a similar question before but didn't get any responses. So I will attempt again (the prior question was regarding Holder continuity). Let $ \Omega$ denote a cube in $ R^n$ and consider …

Suppose $ \Omega$ is a smooth bounded domain in $ R^2$. I am interested in the regularity of solutions to $$-\Delta u(x) = f(x) \mbox{ in } \Omega$$ with $ u=0$ on $ \partial \Omega$. If $ f \in …

I asked a similar question the other day, but I will be more precise now. Consider $ \Omega:=(0,1 ) \times (0,1)$ and consider $$ - u_{xx}(x,y) - u_{yy}(x,y) + a(x) u_x + b(y) u_y + u = f(x,y) \mbox …

I am interested in the regularity of ellitpic equations like $$ -\Delta u(x) +a(x) \cdot \nabla u(x) + C(x) u(x) =f(x) \quad \Omega$$ with $ \partial_\nu u =0$ on $ \partial \Omega$ where $ \Omega= …

Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< \frac{N+2}{N- …