# Tagged Questions

Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

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### Deformation of the covariant Laplacian

Let $M$ be a Riemann surface and $P \to M$ a principal $G$-bundle (with compact structure group $G$). Fix a connection $A$ in $P$ and consider a nearby connection $B$, which is in Coulomb gauge ...
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### elliptic regularity on manifolds

Hello! I'm reading the book of T. Aubin Some nonlinear problems in Riemannian geometry. In chapter 3 he introduces elliptic operators on manifolds, but then he gives regularity results for elliptic ...
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### Linearizing and solving a nonlinear PDE numerically

Im trying to solve the following (transport & diffusion) nonlinear PDE numerically (via finite volume on a cuboid region. Some Material gets cooled down, s.t. in some areas the material becomes ...
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### Manifolds with a lower degree of regularity

I've been reading a paper about regularity theory for a P.D.E in a non-smooth domain(see the reference below). There, the authors consider domains of $R^n$ with regularity of class $W^2 L^{n-1,1}$(...
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### Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}.$

I have a question that it seems simple but I can not solve it. Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...
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Here is the problem: I am trying to figure out if there is a non-zero solution for the system of equations: $(\Delta^{k-1}+a)(d^{\ast}\eta)=2d^{\ast}d\xi$ $(\Delta^{k}+b)(d\xi)=2dd^{\ast}\eta$ for $... 0answers 106 views ### Regularity in PDE theory I stumbled over this question in the context of PDE theory and thought that maybe somebody here knows whether the following is true or not? Let$U$be connected,open and bounded in$\mathbb{R}^n$... 0answers 155 views ### Analytical solution of diffusion PDE with Robin boundary condition I need to find the analytical solution of the time-independent diffusion equation with constant coefficients on the unit disk$\Omega$with subject to Robin boundary conditions. The formulation is as ... 0answers 93 views ### Integrability of$D^2u$for$\infty$-harmonic function$u$? Consider infinity harmonic functions; that is, functions satisfying$\Delta_\infty u = 0$with $$\Delta_\infty u = \langle Du, D^2 u \, Du \rangle = \sum_{i,j} \frac{\partial^2 u}{\partial x_i \, \... 0answers 167 views ### Extra regularity of Poisson problem having nonzero Neumann boundary condition in convex domain Let \Omega\subset\mathbb{R}^2 be a convex simply connected domain having piecewise smooth boundary, f\in L^2(\Omega) and g\in H^{\frac 1 2}(\partial\Omega). Grisvard in [1] among others prove ... 0answers 93 views ### A question about fractional integrals I am starting from Theorem 3.4 in Struwe's book Variational methods. The authors proves an existence result for a problem with critical growth. I would like to replace the laplacian with the ... 0answers 367 views ### problem with non linear pde I have the following pde which i cannot solve. any suggestions, tips on how to approach a solution?$$\left(1-x^3 \frac{\partial y}{\partial x} \partial_{y}\right) f(y(x))-1/4 \left(1-\frac{1}{x^3 (\... 0answers 51 views ### Harmonic functions in tempered distribution sense Suppose$g$is a metric on$\mathbb{R}^3$and$\Omega \subset\subset \mathbb{R}^3$. We assume that$g$is euclidean outside$\Omega$. My question concerns solutions to$\triangle_g u =0$that are say ... 0answers 122 views ### The minimum value of a energy integral Let$D \subset {\mathbb{R}^3}$a simple connected open domain with volume$\int_{\bar D} {dV = 1} $.$\varphi :{\mathbb{R}^3} \to \mathbb{R}$is${C^1}$,$\varphi (\infty ) = 0 $and $${\nabla ^2}\... 0answers 114 views ### How does the L^\infty norm of the solution of -\Delta u + \lambda u =0, \partial_\nu u=\alpha depend upon \alpha and \lambda? Let \lambda > 0 be a constant and let u be the weak solution on a bounded domain \Omega of$$-\Delta u + \lambda u = 0 \quad\text{in$\Omega$}\partial_\nu u = \alpha \quad \text{on$\...
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Let $\Omega$ be a compact, riemannian manifold with non-empty smooth boundary $\partial \Omega = \Gamma$. For a smooth function $u \in C^\infty(\Gamma)$ we define the harmonic extension $\hat{u}$ as ...
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### Concentration compactness on a compact setting

Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\varphi_k \in C^\infty(M)$ such that $\varphi_k$ satisfy the basic concentration compactness ...
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### Hypoelliptic pseudodifferential operators and Fredholm equations?

I read here the following: The parametrix is a useful concept in the study of elliptic differential operators and, more generally, of hypoelliptic pseudodifferential operators with variable ...
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### Localized eigenfunctions of drift Laplacians

I am looking for literature which discusses localization of eigenfunctions of drift Laplacians, i.e. $L\underline u=-\Delta \underline u+\underline{v}.\nabla \underline u$ in 2D/3D domains with ...
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### Sobolev space for manifold with boundary

For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...
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### Schauder-type estimates for polyharmonic operators in a smooth domain of $R^N$

Let $L$ be an elliptic operator of the form $$Lu := (-1)^m \sum_{|\alpha|=2m} a_\alpha(x) D^\alpha u + \sum_{|\alpha|\leq 2m-1} b_\alpha (x) D^\alpha u$$ with smooth coefficients and $u$ defined ...
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### A integral equation with Discrete to result by inverse problem

Problem I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here, (I know the ...
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### Moser's iteration for non homogeneous quasilinear elliptic PDE

I want to know some reference of Moser's iteration for non homogeneous quasilinear uniformly elliptic PDE, in particular, $u_{ij}(\delta_{ij}+u_iu_j|\nabla u|^2)=0$. I want to know how to deal with ...
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### Uniform upper bound for dim of kernel and codimension of range of certain familly of PDE

A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.It is a real analytic vector field on $S^{2}$ which ...
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Let me consider the following subset of probability measures in $R^d$ $$\mathcal{K}_M=\left\{0\leq u(x)\in L^1(R^d):\quad \int u(x)dx =1,\,\int|x|^2u(x)dx\leq M,\,\int u(x)|\log u(x)|dx\leq M\right\} ... 0answers 183 views ### Reference request: optimal L^p regularity for solutions to -\Delta u=f with f\in L^1(R^d) The tilte says it all. Given f\in L^1(R^d) (let me restrict to dimension d\geq 3 for convenience), what is the optimal L^p regularity for solutions to$$ -\Delta u=f\hspace{3cm}(1)? $$I'm of ... 0answers 51 views ### reference on existence result for nonlinear elliptic PDE During my work, I came to the question of existence of weak solutions to the following elliptic equation$$\triangle u + \partial_{1} u + \partial_2(f(x_1,x_2,u)) = 0 \hbox{ in } \Omega$$with ... 0answers 103 views ### slightly subcritical elliptic pde; the linearized equations Let  p_m \nearrow \frac{N+2}{N-2} and consider the family of elliptic problems$$-\Delta u_m(x)=u_m(x)^{p_m} \quad B \qquad \quad u_m =0 \quad \partial B,$$where B is the unit ball ... 0answers 90 views ### A parametrix for the \bar\partial operator adapted to a holomorphic foliation Let X be a compact complex manifold with (regular) holomorphic foliation given by a holomorphic subbundle \mathcal F of the tangent bundle. The foliation induces a filtration on differential forms.... 0answers 102 views ### Regularity of solution of nonlinear equation Hi! Let L be a linear elliptic operator of order 4 with smooth and bounded coefficients on the ball B_1 of R^{2n} and let N\in C_{loc}^{0,\alpha}(R^{3}). Let f\in C^{0,\alpha}(B_1) ... 0answers 197 views ### Core of divergence form operator with unbounded coefficient Consider the unbounded operator L on L^2(\mathbb{R^d}) to be the self-adjoint extension of$$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$on C^2_c(\mathbb{R^d}). I also assume that a(x)... 0answers 175 views ### A first order ODE with Sobolev estimates Suppose f: {\mathbb R}\to {\mathbb R} is a continuous function satisfying$$f(s) = c_- >0,\forall s<-T; f(s) = c_+<0, \forall s>T. (For simplicity we may even assume that $f$ is smooth,...
I wanna discretize the following free boundary problem Find $u$ and $\Omega$ such that $\Delta u=1-\delta_0$ in $\Omega$ with the conditions $u=|\nabla u|=0$ on $\partial \Omega$. I apply finite ...