All Questions
Tagged with fa.functional-analysis elliptic-pde
230 questions
4
votes
1
answer
343
views
Conditions for the existence of a solution to a semilinear second-order PDE with a-priori bounds
Consider the general semilinear elliptic second-order PDE
$$
u_t-\mathcal L u=f\left(t,x,u,\nabla u\right)
$$
where $\mathcal L$ is an elliptic linear operator (like minus the Laplace operator), $t \...
- 91
3
votes
0
answers
324
views
Would you help me to find this expression?
I'm studying a paper, which I'll include a small piece here. And I'm struggling to calculate
$$C_n\|u_{m,n}\|^{\left(\frac{2*}{2}\right)^k\frac{2*-q}{(r_k)^k}}_{L^{2*}(\Omega)}$$
Where $\Omega$ is an ...
- 63
1
vote
0
answers
74
views
Reference request: normal trace and the conormal derivative associated to the operator $Div (A \nabla)$ for a symmetric positive definite $A$
Let $A$ be a $3\times 3$ symmetric positive definite matrix. I am looking for a reference where I could find in which sense the normal trace $\gamma$ and conormal derivative $\gamma_n$ associated to ...
- 63
2
votes
1
answer
129
views
Spectral analysis for nonlocal elliptic operator
Suppose $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary. We note by $(-\Delta)^{-1}$ the inverse Laplacian i.e. $f\mapsto u$ where $u$ is the unique solution to
$$-\Delta u=f,\...
- 287
3
votes
0
answers
170
views
A version of the Nash-Moser inverse function for unbounded domains?
Do there exist versions of the Nash-Moser inverse function theorem applicable to spaces of unbounded smooth functions on unbounded domains in $\mathbb{R}^n$? Any reference would be appreciated but ...
- 505
9
votes
2
answers
1k
views
Density of restrictions of harmonic functions inside a ball
Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let
$$ S_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text{...
- 4,135
2
votes
1
answer
432
views
Second-order elliptic regularity with rough coefficients
Let $\Omega \subseteq \mathbb R^n$ be a bounded open set with smooth boundary. Let $k\geq 1$, $\alpha\in(0,1)$, $a_{ij},b_i,f \in C^{k,\alpha}(\Omega)$ for $i,j=1,...,n$, and define the operator
$$L = ...
- 45
1
vote
0
answers
94
views
Existence of $C^{2, \alpha}$ solution to $a^{ij}(x,u,Du)D_{ij}u+b(x,u,Du)=0$ using the Leray–Schauder theorem in "Elliptic PDE" of Q. Han & F. Lin
In this part of the book "Elliptic PDE" of Qing Han & Fanghua Lin, the Leray–Schauder existence theorem is applied to prove the existence of $C^{2, \alpha}(\bar{\Omega})$ solution.
For $\...
- 809
0
votes
1
answer
239
views
A proof for the existence of smooth solution of PDE in form $\Delta u=f(x, u)$ in Michael E. Taylor's book Partial Differential Equations III
This part is from page 107 in Michael E. Taylor's book Partial Differential Equations III.
In this part, we want a proof for the existence of smooth solution of the PDE
$\Delta u=f(x, u)$ on $U$ with ...
- 809
2
votes
1
answer
160
views
Elliptic equation on square with periodic boundary values for the solution and it's partial derivatives
Suppose that $\Omega=[0,1]^2$. I will say that a real valued function $u$ on $\Omega$ satisfies periodic boundary values if
$$u(x,0)=u(x,1), \;u(0,y)=u(1,y),\;\;\;\text{ for all }x,y\in[0,1].$$
Now ...
- 153
0
votes
1
answer
162
views
Iterated integrations by parts using the fractional Laplacian
Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that
$$\...
- 839
4
votes
1
answer
285
views
Elliptic regularity when the Lagrangian is possibly infinite
I want to solve variational problems of the form
$$\inf_u \int_{-1}^1 \phi(u'(x)) \text{ with } u(-1)=u(1) = 0,$$
where $\phi(p)$ is convex and is allowed to take on the value $+\infty$ for some ...
- 981
2
votes
1
answer
246
views
Change of variable and boundary data for Laplace equation
Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem
$$
\begin{cases}
-\Delta u = 0 & x \in \Omega \\
u = 1 & x \in \partial \Omega
\end{cases}
$$
Does it make ...
- 161
2
votes
1
answer
196
views
Viscosity solutions of $(-\Delta)^s u = 0$ in $\Omega $ with non-homogeneous data $u = 1$ in $\mathbb R^n \setminus \Omega$
Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem
$$
(1) \quad \begin{cases}
(-\Delta)^s u +\lambda u= 0 & x \in \Omega \\
u = 1 & x \in \mathbb R^n \...
- 161
7
votes
1
answer
339
views
Does the pointwise mean value property imply harmonicity?
Assume $u:\Omega\subset\mathbb{R}^d\to\mathbb{R}$ is continuous and satisfies the property:
for every $x\in \mathbb{\Omega}$ there is $r_x>0$ such that
$$
u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(...
- 1,101
4
votes
0
answers
174
views
Techniques for showing non-degeneracy results (PDE)
Motivation:
Consider the equation,
$$-\Delta u = u^p$$
in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,...
- 537
7
votes
2
answers
508
views
A little problem in PDE or function analysis
Let
$E$ be the usual sobolev space $H^{1}_{0}(\Omega)$ on a smoothly bounded domain $\Omega$,
$E_{k}$ be its subspace spanned by the first $k$ eigenfunctions of the Laplace operator, i.e.
$$E_{k}:=\...
- 705
1
vote
0
answers
130
views
Results on the eigenspace of weighted elliptic eigenvalue problems
I am considering the following eigenvalue problem in $\Omega=\mathbb{R}^n_+$
$$-\operatorname{div}(a(x)\nabla \varphi) = \lambda a(x)w(x)\varphi$$
where the weights $a>0$ and $w\in L^{\infty}$ (and ...
- 537
2
votes
1
answer
173
views
Semi-linear elliptic problem, energy functionals, Fréchet derivatives and the Newton method in Banach spaces
Suppose $\Omega\subset\mathbb{R}^n$ is a regular open set, $f\in L^2(\Omega)$ and consider the following elliptic problem.
$$-\Delta u + u=f'(u) , \;\;u_{|\partial \Omega}=0,$$
where $f'$ is the ...
- 597
2
votes
2
answers
132
views
Density of traces of solutions to an elliptic equation
Let $D_1$ be a domain with smooth boundary and assume that $D_1$ is a proper subset of $D_2$ which is itself a bounded domain in $\mathbb R^n$ with a smooth boundary. Assume also that $D_2\setminus ...
- 4,135
1
vote
1
answer
182
views
Proving an estimate for the Neumann problem on $\mathbb{R}^3 \setminus B_1$ in Weighted Sobolev spaces
Let $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
It is known that for every $g \in H^{\frac{1}{2}} (\partial M)$, and for an appropriate $\delta$, there exists a unique solution $u$ ...
- 969
1
vote
0
answers
298
views
Is there a generalization of the Agmon-Douglis-Nirenberg regularity theorem for elliptic equations to domains with corners?
The Agmon-Douglis-Nirenberg theorem(s) state(s) that whenever $f\in W^{m,p}(\Omega)$ where $\Omega$ is a bounded open set of class $C^{m+2}$, then there is a unique solution $u\in W^{m+2,p}(\Omega)$ ...
- 597
5
votes
2
answers
1k
views
How to prove the second Korn inequality?
$\textbf{Theorem}.1$ (The first Korn inequality) Suppose that $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ with Lipschitz boundary. Then\
\begin{eqnarray}
\sqrt{2}\left\|\triangledown u\right\|_{...
- 1,219
2
votes
0
answers
94
views
From some priori estimates can we estimate higher Sobolev norm?
Suppose $u$ is a smooth function on bounded set $\Omega$ with smooth boundary such that
$$\|u\|_{W^{1,p}(\Omega)}\le C\|\phi\|_{W^{1-1/p,p}(\partial\Omega)}$$
where $u|_{\partial\Omega}=\phi$.
Can we ...
- 309
1
vote
0
answers
103
views
Regularity results for non uniform elliptic equation
I have seen some regularity result for ellptic PDE but all of them consist of uniform elliptic one. For instance,
$$\nabla \cdot (\gamma(x) \nabla u)=F \text{ in } \Omega\qquad u=\phi \text{ on }\...
- 309
3
votes
0
answers
96
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
$$\begin{cases} -\Delta ...
- 969
2
votes
0
answers
147
views
Dimension of critical set of p-harmonic function
Let $\Omega\subset \mathbb{R}^n$ be a smooth domain and $u\in W^{1,p}(\Omega)$ a non-constant $p$-harmonic function, for some $1<p<n$.
Question: What is the Hausdorff dimension of the critical ...
- 91
2
votes
0
answers
125
views
Regularity up to boundary of a solution $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ to $\Delta^2 u = -\text{div}\, F$
Let $\Omega\subset \Bbb R^n$ be a $C^{2}$ domain (open and bounded) and let $p\in(1,\infty)$. Suppose $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ is a weak solution to the fourth-order elliptic system
...
- 1,245
4
votes
1
answer
698
views
Poincare Inequality for $H^2$ function satisfying homogeneous Robin boundary conditions
Let $\Omega\subset\mathbb{R}^3$ be a bounded smooth domain. In general, for a Poincare inequality of the type
$$\|u\|_{L^2}\le C \|\nabla u\|_{L^2}$$
to hold for all $u\in X\subset H^1(\Omega)$ and $C$...
- 287
4
votes
0
answers
143
views
If theorem valid for compactly supported distribution then is it also valid for tempered distribution?
I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution.
For instance,
Theorem: Any $A \in \Psi^{m}$ ...
- 309
0
votes
1
answer
98
views
Reference request: Is if possible to estimate the local behaviour of the solution of $\nabla \cdot a(x) \nabla f=g$ via constant coefficients?
Consider the divergence form uniformly elliptic operator $\nabla \cdot a(x) \nabla$
where the coefficient $a$ are smooth and bounded and $D$ is a bounded
and smooth domain of $\mathbb R^d$
$$
\begin{...
- 446
2
votes
1
answer
287
views
Reference request for semilinear PDEs in dimension 2
I am interested in the study of the (semi-linear, I suppose) equation
$$\begin{cases}-\Delta u(x,y)+q(x)u(x,y)+h(x)=f(u(x,y)-kx),\;\;(x,y)\in\Omega,\\
u=g,\;\;\;\text{on }\partial\Omega.\end{cases}$$
...
- 597
5
votes
1
answer
188
views
Does a suitable famlly of eigenvectors of non self-adjoint operators, sufficiently close to an adjoint one, form a basis?
It is very well known that if $A\in L^\infty(B_1;\mathbb R ^{d\times d})$ is a positive definite symmetric matrix, the eigenvalue of the self adjoint operator $H^2(B_1)\cap H^1_0(B_1)\to L^2(B_1)$
$$T:...
- 2,494
1
vote
0
answers
105
views
Derivative and Green function of Fractional Laplacian in a bounded domain: $(-\Delta)^s\nabla_x G(\bar x,z) = 0 \text{ in } \Omega $?
Let $G$ be the Green function of the Fractional Laplacian $(-\Delta)^s$ in a domain $\Omega$ (which is known explicitly for the special case of the ball: link). Let $\bar x \in \Omega$ be fixed. Does ...
- 303
3
votes
1
answer
333
views
References for Green functions of $\nabla \cdot a \nabla$ on a domain with $a \in L^\infty$
I am looking for a reference for basic properties of the Green function for a symmetric, uniformly elliptic operator $\nabla \cdot a \nabla$ where the coefficients $a_{ij}= a_{ji}$ are only assumed to ...
- 446
1
vote
1
answer
274
views
Poisson equation in a periodic strip
Consider the periodic strip $\Omega=\mathbb{T}\times[0,1]$ where $\mathbb{T}$ is the 1D torus with period 1. We consider the mixed Dirichlet/Neumann problem
$$-\Delta u=f$$
with boundary conditions
$$...
- 287
0
votes
0
answers
150
views
Eigenvalues of the Laplacian and min-max formula in any space dimension
In which reference book can I find a proof that the eigenvalue of the Laplace operator in a domain $\Omega \subset \mathbb R^d$ with $d \ge 1$ are given by
$$
\lambda_1 = \min_{u \in H^1_0(\Omega), \|...
- 99
1
vote
0
answers
144
views
Liouville theorem for elliptic equation with advection term
How can one prove that any $L^2$ solution of
$$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } \mathbb R^N $$
is zero if $a(x)$ is a divergence-free vector field such that
$\int |\...
- 99
1
vote
0
answers
130
views
Fractional Sobolev embedding theorem
Let $\psi \in C^\infty_c(\mathbb R^N)$ be a test function with support iN $B(0,R)$. Is it true that the following inequality holds
$$\int_{B(0,R)} \psi^2 u^{\frac{4}{1+\beta}} dx \le R^{1+\beta} \int_{...
- 99
1
vote
1
answer
187
views
Fractional Laplacian equation on a ball and explicit solutions
Let us consider
\begin{align*}
(-\Delta)^s u &= 0 && x \in B_r(0) \\
u&=0 && x \in \mathbb R^N \setminus B_r(0),
\end{align*}
where $$
(-\Delta)^s u(x) = \int_{\mathbb{R}^N} \...
- 99
1
vote
1
answer
195
views
Existence and regularity for fractional elliptic problem with gradient term: $ (-\Delta)^s u + v\cdot \nabla u = 0$ with $v \in \dot H^s$
Let us consider the problem
$$ (-\Delta)^s u + v\cdot \nabla u = 0 \quad \text{ in } \mathbb R^n, $$
where $s \in (0,1)$, $(-\Delta)^s$ is the fractional Laplace operator and
$v:\mathbb R^n \to \...
- 99
2
votes
0
answers
52
views
Reference Request: Dirichlet operators with singular coefficients
Let $d\geq 2$, $\delta \in (0,1)$ and let $\mathcal{L}_{d,\delta}$ be the second order differential operator defined by
\begin{align*}
\mathcal{L}_{d,\delta}(f)(x) = \Delta(f)(x)-\delta \|x\|^{\delta-...
- 778
1
vote
0
answers
122
views
Series and solution of $-\Delta u + \lambda u = f(x)$
Consider a bounded smooth set $\Omega \subset \mathbb R^n$ (for example, we can take a ball). Can we write down the solution of
\begin{align*}
-\Delta u(x) + \lambda u(x) &= f(x), & x \in \...
- 131
3
votes
1
answer
362
views
Solution of the fractional Laplace equation on a ball
What is the expression of the (non $u \equiv 0$) solutions to
\begin{align*}
(-\Delta)^s u &= 0 && x \in B_r(0) \\
u&=0 && x \in \mathbb R^N \setminus B_r(0),
\end{align*}
...
- 99
-1
votes
1
answer
102
views
Is it true that $\nabla_x \int_0^\infty f(t,0) dt = 0 \implies \nabla_x f(t,0) = 0 \ \forall t>0$? [closed]
Let $f:\mathbb R_+ \times \mathbb R^N \to \mathbb R$ and $$F(x) = \int_0^\infty f(t,x) dt.$$ If $\nabla_x F(0) = 0$ do we have that $\nabla_x f(t,0) = 0$ for all $t \in \mathbb R_+$? If not, which ...
- 131
0
votes
2
answers
238
views
Fractional Laplacian of $(a-x)_+^\alpha$ in $(0,1)$
How can I compute the spectral fractional Laplacian of $(a-x)_+^\alpha$ on $\Omega = (0,1)$?
Here the operator is defined as $$(-\Delta)^s u = c_{N,s} \int_0^\infty (e^{t\Delta_N}u(x) - u(x)) t^{-1 - ...
- 109
0
votes
1
answer
344
views
Is this PDE solvable?
Let $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is unit ball. I am trying to solve the following PDE for $f$:
$$\Delta f -\frac{ f }{r^2}+ \frac{ \left. f \right|_{\partial M}}{r^2} = 0, \qquad \text{...
- 969
0
votes
1
answer
114
views
Bounding $(\int_{S^1}\left|\partial_r u(r\omega)\right|^2 d\omega)^{1/2}$ with $(\iint \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy)^{1/2} $?
The following inequality is trivially true
$$\left(\int_{S^1}\left|\frac{\partial u}{\partial r}(r\omega)\right|^2 d\omega\right)^{1/2} \le \left(\int_{S^1}\left|\nabla u(r\omega)\right|^2 d\omega\...
- 161
5
votes
1
answer
297
views
A scaled fractional ''Sobolev inequality''
Does a fractional interpolation inequality similar to $$
\int_{B_R(0)} |u| dx \le C R^{2} \sqrt{\log(2R)} \Big( \int_{\mathbb R^2}\int_{\mathbb R^2} \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy + \int_{B_1(...
- 109
7
votes
0
answers
351
views
Fractional Laplacian and chain rule
For the classical Laplacian, we have
$$\Delta (h(u)) = h'\Delta u + h''(u)|\nabla u|^2$$
for smooth functions $h$ and $u$.
Does a similar chain rule hold (up to a reminder term) also for the ...
- 161