Questions tagged [elliptic-pde]
Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
518
questions with no upvoted or accepted answers
3
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answers
63
views
Existence of ground state solutions for the critical exponent
I have been recently reading Kwong's paper on the uniqueness of positive solutions for the equation $\Delta u-u+u^p=0$ in $\mathbb{R}^n$.
The authors show that the above equation has a unique positive ...
3
votes
0
answers
151
views
A version of the Nash-Moser inverse function for unbounded domains?
Do there exist versions of the Nash-Moser inverse function theorem applicable to spaces of unbounded smooth functions on unbounded domains in $\mathbb{R}^n$? Any reference would be appreciated but ...
3
votes
0
answers
63
views
Eigenvalues of an elliptic operator on shrinking domains
This was probably done somewhere 100 times, but I can't find a reference.
Assume that we have a bounded star-shaped domain $\Omega\subset \mathbb{R}^n$ with piece-wise smooth boundary and a general ...
3
votes
0
answers
356
views
Regularity of solution to Laplacian equation with Neumann data on Lipschitz domain
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and let $u\in H^1(\Omega)$ be a weak solution to
\begin{equation}
\begin{cases}
-\Delta u=0 \quad &\mbox{in $\Omega$}\\
\frac{\partial ...
3
votes
0
answers
191
views
The regularity of solutions to the Neumann problem for an elliptic PDE on a domain with piecewise smooth boundary
While doing my research, I encountered the following problem as:
is there any regularity result for solutions to the Neumann problem for an elliptic PDE on a domain with piecewise smooth boundary? For ...
3
votes
0
answers
95
views
A sequence of functions solving $-\Delta u_n + V u_n = u_{n-1}|_{\partial M}$
Let $M = \mathbb R^3 \setminus B_1$ where $B_1$ is the unit ball.
Let $ h \in C^{\infty}(\partial M)$ and let $u_0$ be the unique function that vanishes at infinity and solves
$$\begin{cases} -\Delta ...
3
votes
0
answers
83
views
Minimal normal graph
Let $(M^3,g)$ be a complete and orientable Riemannian $3$-manifold and let $\Sigma^2 \subset M$ be a compact orientable totally geodesic surface embedded in $M$. For $f \in C^{2,\alpha}(\Sigma)$ with ...
3
votes
0
answers
131
views
Homogeneous Carnot group, its Lie algebra and Carnot-Carathéodory ball
Background: Let the smooth vector fields $X=(X_1,\cdots,X_m)$ define on $\mathbb{R}^n$ and they satisfy the following assumption:
(H1): There is a dilation structure
$$\delta_{t}:\mathbb{R}^n\to \...
3
votes
0
answers
196
views
integration by parts on a Lipschitz domain as $\epsilon\to 0$
For a fixed, bounded, smooth domain $\Omega\subset \mathbb R^d$ and any $u\in W^{1,1}(\Omega)$ with trace $u|_{\partial\Omega}=g\in L^1(\partial\Omega)$ one can prove that
$$
\lim\limits_{\epsilon\to ...
3
votes
0
answers
270
views
Integral representation of solution of an elliptic PDE in divergence form
Suppose we have a second order elliptic differential operator
$$
L(v) = -\text{div}(A(x) \nabla v)
$$
$A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And ...
3
votes
0
answers
185
views
Regularity of harmonic functions for a degenerate elliptic operator
This is a question on a degenerate elliptic operator.
Let $E$ be a closed unit ball in $\mathbb{R}^d$ centered at the origin. For a positive number $c>0$ and $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...
3
votes
0
answers
76
views
Reference requests: $W_{p}^1$-estimate for $(\triangle -\lambda)$ on Lipschitz domains
Let $1<p<\infty$ and $\lambda>0$. When $\Omega$ is a bounded $C^1$ or a bounded Lipschitz domain
with small Lipschitz constant in $\mathbb{R}^d$, then for every $f\in L_p(\Omega)$ and $\...
3
votes
0
answers
64
views
Elliptic equations in semi-infinite strips
Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary. Let $g=(g_{jk})_{j,k=0}^n$ be a Riemannian metric on $\mathbb R^+\times \Omega$ with smooth bounded components. Is there a good ...
3
votes
0
answers
214
views
Does the minimal surface system in the plane have the weak unique continuation property?
Let $\Omega \subset \mathbf{R}^2$ be a domain in the plane and suppose that $u : \Omega \to \mathbf{R}^k$ is a smooth function for which the graph of $u$ is a smooth minimal surface in $\Omega \times \...
3
votes
0
answers
183
views
How to prove the following linearized operator is positive?
In $L^2(\mathbb{R}^d)$, let $Q$ be the solution to
\begin{equation}
-\Delta Q+\alpha^2 Q = |Q|^{2\sigma} Q,
\end{equation}
and $Q$ satisfies that it is positive, radial, and exponentially decaying (...
3
votes
0
answers
55
views
Harnack inequality for $F(D^{2}u, Du,x)=f$
I would like to know if there is an harnack inequality for the viscosity solutions of the following pde in the literature: $F(D^{2}u,Du,x)=f \in L^{\infty}(B_{1})$, for a function $u \in C(B_{1})$, ...
3
votes
0
answers
105
views
Complex Monge-Ampere equation with degenerate right hand side
Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation:
$(\omega_0 +i \partial \bar \partial \varphi)^...
3
votes
0
answers
119
views
Second derivative estimates
I am in big trouble since I don't see how to proceed (I don't need the exact calculation) with the following estimates.
In one of his papers, Lin proves the following result:
Let's consider a ...
3
votes
0
answers
94
views
About p-laplacian and variations
Let $\Omega \subset \mathbb{R^{n}}$ be a domain (open and connected set), for $p\geq 2$, the $p$-laplacian is defined by:
$\Delta_p u= \operatorname{div} (|\nabla u|^{p-2} \nabla u)$, in non-...
3
votes
0
answers
95
views
Partial regularity of harmonic maps into spheres
Let $u : B_1(0) \subset \mathbb{R}^n \to S^k$ be an energy minimizing (minimizing in $H^1(B_1(0); S^k)$) harmonic map. I am trying to understand the theory of partial regularity, where the main claim ...
3
votes
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answers
191
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Singular integral operators and PDEs
What is the relation between the notion of singular integral operators and partial differential equations?
I know, for example, that there is a relation between the Cauchy transform (Riesz transforms ...
3
votes
0
answers
86
views
Parabolic regularity for the 2D Navier-Stokes equations in a bounded domain
Suppose we consider 2D Navier-Stokes equations in a bounded domain $\Omega \subseteq \mathbb R^2$, together with suitable boundary conditions so that we can consider the vorticity equation: $$\omega_t ...
3
votes
0
answers
67
views
Diffusion generators with gradient vector fields
Let $\mathcal{A}$ be a second order operator on an $n$-dimensional smooth manifold $M$, expressed in Hörmander form as
$$\mathcal{A}=X_0+\frac{1}{2}\sum_i^kX_i^2,$$
where $X_0,X_1,...,X_k$ are ...
3
votes
0
answers
127
views
Unique continuation from the boundary for inhomogeneous elliptic pde
Let $Lu = f$ be satisfied on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$, where $L$ is a strongly elliptic second order differential operator with real ...
3
votes
0
answers
162
views
Asymptotic behaviour of principal eigenfunctions and large deviations
Dear Math Overflowers,
I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more ...
3
votes
0
answers
105
views
metric with curvature bounded in $L^2$
My question is about the regularity of a metric whose curvature is bounded in $L^2$. Of course, this question doesn't really make sense since the regularity of the metric depends on the coordinates ...
3
votes
0
answers
112
views
Parametrix of external product of elliptic operators
Let $S$ and $T$ be Dirac-type operators on vector bundles $E\rightarrow M$ and $F\rightarrow N$ respectively, where $M$ and $N$ are manifolds. Suppose $Q_S$ and $Q_T$ are parametrices for $S$ and $T$. ...
3
votes
0
answers
79
views
Generalized viscosity sub(super)solution and it's convolution
Suppose that $\Gamma \subsetneq \mathbb{R}^n$ is an open symmetric convex cone containing positive orthant.
Note that $\Gamma \subset \left\{x=(x_1,...,x_n) \in \mathbb{R}^n | \sum_{i=1}^{n} x_i > ...
3
votes
0
answers
84
views
Estimate a function given an estimate of its Laplacian
Let $f_\lambda\geq 0$ with $\lambda>0$, be smooth functions in the unit Euclidean ball $B\subset \mathbb{R}^n$ satisfying the following conditions:
\begin{eqnarray*}
\int_B |f_\lambda(x)|^2dx\leq 1,...
3
votes
0
answers
73
views
Finding the particular and general solutions to Einstein Field Equations under generalized Vaidya Geometry
The problem I have is on finding the particular and general solutions to Einstein Field Equations under generalized Vaidya Geometry, which comes from the following paper : https://journals.aps.org/prd/...
3
votes
0
answers
120
views
Partial regularity for transmission problem in corner domains
Let $n=2$ or $3$ and $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Let suppose that $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and we define $\Gamma = \partial \Omega_1 ...
3
votes
0
answers
182
views
Families of unbounded operators
Let $H$ be a Hilbert space, $X$ a topological space, and $\{A_t\}_{t\in X}$ a continuous family of bounded, invertible operators on $H$. Continuous here in the sense that the corresponding map $X\...
3
votes
0
answers
72
views
Elliptic operator applied to the distance function
Let $\Omega$ and open subset of $\mathbb{R}^n$. Let us consider the following operator:
$$
\Delta_A (u)\, \, \colon= \text{div}(A \nabla u ), \qquad u \in C^{\infty}(\Omega)
$$
where $A(x)$ is a ...
3
votes
0
answers
145
views
When a PDE add a Laplacian term
I went to a talk today and the speaker mentioned when you add a Laplacian term to a PDE, the Laplacian will dominate (in what sense?), which I don't quite understand. I know this question is a bit ...
3
votes
0
answers
260
views
Principal eigenvalue of Laplacian under volume preserving mean curvature flow
Consider a compact uniformly convex n-dimensional hypersurface $M_0$ without boundary , which is smoothly imbedded in $\mathbb R^{n+1}$ , and suppose that $M_0$ is represented locally by some ...
3
votes
0
answers
141
views
Prove the positivity of the subelliptic operator heat kernel
Let $X=(X_{1},X_{2},\cdots,X_{m})$ be $C^{\infty}$ real vector fields defined in $\mathbb{R}^{n} (n>2)$ satisfying the Hormander's condition at every point $x\in\mathbb{R}^{n}$, Let $W$ be a ...
3
votes
0
answers
553
views
2D laplace equation with Robin boundary condition (Green function)
Let's say that I know a fundamental solution for the Laplace equation in the whole plane:
$$\nabla^2u=\delta\quad \text{in the sense of distributions,}$$
and I need a solution for the laplace equation ...
3
votes
0
answers
179
views
Characterization of minimizer of convex functional
I need to check whether the following characterization of the minimizer of a convex functional is valid.
Let $X$ be a reflexive Banach space (think $W^{1,p}(\Omega)$ with $\Omega \subset \mathbb R^n$ ...
3
votes
0
answers
124
views
Wave equation that becomes elliptic on a bounded domain (sign-changing coefficient)
I'm looking for results on this kind of problems:
$$ \partial_{tt}^2 u - \partial_x(a(x) \partial_x) = f,$$
$$u(t=0) = u_0, \quad \partial_t u(t=0) = u_1,$$
where $a$ changes sign: $a(x)= -c^2 < 0$ ...
3
votes
0
answers
133
views
Sufficient condition for the unique solvability of Dirichlet problem of Hamilton-Jacobi equation
It shall be an old story in PDE.
I am looking for a sufficient condition of Dirichlet problem for the existence of the unique viscosity solution of the equation in the form of
$$\inf_{a \in [-1,1]} \{...
3
votes
0
answers
107
views
A priori $C^0$ estimates for a semi-linear vector Poisson equation
Main Question
Consider a $C^2,H^2$ map $F:\mathbb{R}^m \to \mathbb{C}^n$ which satisfies the following equation:
$$
-\Delta F(x) + \sum_i a_i(x)\nabla_iF(x) + B(x)F(x) + |F(x)|^2F(x) = 0
$$
Here $a_i:...
3
votes
0
answers
73
views
On the principal eigenvector of an elliptic operator
Suppose I have an open domain $U \subset \mathbb{R}^n$ and an elliptic operator $L$ acting on all square-integrable $C^2$ functions $\rho:U\to \mathbb{R}$ which converge to zero at $\partial U$:
\...
3
votes
0
answers
244
views
Interior regularity for elliptic operators with non smooth coefficients
I need a pretty standard interior regularity result for a second order elliptic operator of the form
$$
-\nabla^b \cdot (A(x) \nabla^b v)+c v=f,
\qquad
\nabla^b=\nabla+ib(x)
$$
where $A(x)$ is a ...
3
votes
0
answers
134
views
Equations of the form $((\Delta)^{2}+\lambda\Delta+\gamma)(f)=0$
Here is the problem: I am trying to figure out if there is a non-zero solution for the system of equations:
$(\Delta^{k-1}+a)(d^{\ast}\eta)=2d^{\ast}d\xi$
$(\Delta^{k}+b)(d\xi)=2dd^{\ast}\eta$
for $...
3
votes
0
answers
127
views
Existence of at least one positive solution for semilinear biharmonic equation with critical exponent
Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow
\begin{cases} \Delta^2u=\lambda \dfrac{u}{|x|^4}+u^{p} & \mathrm{in}\hspace{...
3
votes
0
answers
161
views
probabilistic interpretation of elliptic equation with mixed boundary condition
I would like to understand the probabilistic interpretation of the following elliptic problem with mixed Dirichlet-Neumann boundary conditions:
Let $B := \{ x \in \mathbb{R}^n, \quad \| x \|_2 \leq 1 ...
3
votes
0
answers
324
views
Extra regularity of Poisson problem having nonzero Neumann boundary condition in convex domain
Let $\Omega\subset\mathbb{R}^2$ be a convex simply connected domain having piecewise smooth boundary, $f\in L^2(\Omega)$ and $g\in H^{\frac 1 2}(\partial\Omega)$. Grisvard in [1] among others prove ...
3
votes
0
answers
118
views
A question about fractional integrals
I am starting from Theorem 3.4 in Struwe's book Variational methods. The authors proves an existence result for a problem with critical growth. I would like to replace the laplacian with the ...
3
votes
0
answers
263
views
Result like Brezis-Kato Lemma for Biharmonic equation.
Is there any result for the 4th order elliptic pde,like we do have for 2nd order pde: Consider the equation,\ $\Delta^2 u = a(x)u$ and $a(x)\in L^{\frac{n}{4}}(\Omega)$, where $\Omega$ is bounded. If $...
3
votes
0
answers
379
views
problem with non linear pde
I have the following pde which i cannot solve. any suggestions, tips on how to approach a solution?
$$\left(1-x^3 \frac{\partial y}{\partial x} \partial_{y}\right) f(y(x))-1/4 \left(1-\frac{1}{x^3 (\...