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Questions tagged [elliptic-equations]

For questions about elliptic partial differential equations. If your question is specific to the Laplace equation, see (harmonic-functions).

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what is called this sub space $ H^{1}_{0,p} $

What's the name of this sub sobolev space $ H^{1}_{0,p}= \left\{ u\in AC\left( \left[0,+\infty \right), \mathbb{R} \right) : u\left( 0 \right) = u\left( +\infty \right) =0 , \sqrt{p} \acute{u} \in ...
Daking Diss's user avatar
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1 answer
37 views

Least Squares Ellipse with known parameters

Given a set of points in 2D space $$P = \{(x_i, y_i), \text{for } i \text{ in }1 \dots N\}$$ I want to find the least squares fit of an ellipse $$\frac{(x - c_x)^2}{r_x^2} + \frac{(y - c_y)^2}{r_y^2} =...
Dominik Ficek's user avatar
-2 votes
0 answers
20 views

regularity of the weak solution on the cube [closed]

Let $Q:=[0,1]^d$ and $g\in L^2(\Omega)$ consider the PDE : $$ \left\{ \begin{array}{ll} -\Delta f=g & \text{in $Q$} \\ f\equiv 0 & \mbox{on $\partial Q$}, \end{array} \...
Alucard-o Ming's user avatar
1 vote
0 answers
30 views

Problem 2.1 on page 3 in Avner Friedman's PDE textbook.

For notation look at this link here: Problem 1 on page 2 of Avner Friedman's PDE book. They ask the following: Let $$L=\sum_{|\alpha |\le m} a_{\alpha}(x)D^{\alpha}\equiv \sum_{k=0}^m\sum_{\...
MathematicalPhysicist's user avatar
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0 answers
39 views

Why is there a focus on positive solutions of elliptic eigenvalue problems?

I am looking at the theory of principal eigenvalue problems of elliptic operators. A simple form would be \begin{align} \label{eu_eqn} Lu(x) &= \lambda u(x) \quad x \in \Omega \\ u(x) &= 0 \...
Ozymandias's user avatar
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1 answer
41 views

Nonhomogeneus PDE function requires expansion in sines?

I'm studying about the solution to the PDE: $$ \Delta u(x,y)=-f(x,y)\\ u(0,y)=u(a,y)=0 \\ u(x,0)=g(x) \ \ , \ \ u(x,b)=h(x) $$ And the first step is to start solving it like a homogeneus equation with ...
Krum Kutsarov's user avatar
1 vote
0 answers
26 views

Regularity of an elliptical problem

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain with boundary of class $C^{\infty}$, with $N \geq 3$. Moreover, consider the uniformly elliptic operator $$\mathcal{L}v := -\Delta v - \lambda v.$$...
Santos's user avatar
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Elliptic boundary value problem with time dependency

I am looking at an elliptic boundary value problem for an open set $\Omega\subset \mathbb{R}^3$ that is solved over a time interval $(0,T)$ with $T>0$ \begin{equation*} \begin{cases} \begin{aligned}...
Nik's user avatar
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1 vote
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Compact inverse of elliptic operator

In the book of Schoen-Yau, "Lectures on Differential Geometry", in Proposition 1.1, page 188, they said that: Define $Lu= - \Delta u + cu$. It is well known that, as an elliptic operator ...
Duc's user avatar
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3 votes
1 answer
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Problem 1 on page 2 of Avner Friedman's PDE book.

First some notation. $|\alpha|=\alpha_1+\ldots +\alpha_n$, $x=(x_1,\ldots,x_n)$, $D_i=\frac{\partial}{\partial x_i}$, $D^\alpha=D_1^{\alpha_1}\ldots D_n^{\alpha_n}$. A linear partial differential ...
MathematicalPhysicist's user avatar
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13 views

Continuous dependence of the solution on the data constant

Considering $Lu=f$ to be a linear symmetric self-adjoint problem with L being a differential operator of order $2m$, the book finite element book by Strang and Fix claims that $$ ||u||_{2m} \leq C ||f|...
mz _'s user avatar
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Having trouble proving an inequality

I was confused about a step when looking at a simple transformation of an inequality in a book Elliptic Partial Differential Equations by Qing Han and FangHua Lin. The picture below is the complete ...
PauseAndPonder's user avatar
4 votes
3 answers
110 views

Dense subspace of space of radial functions in $H^1_0(\Omega)$

Consider $\Omega\subset \mathbb{R}^N$ a bounded domain, which is radial. By definition, the space of radial functions in $H^1_0(\Omega)$ is $$ H^1_{0, rad}(\Omega) = \{u \in H^1_0(\Omega) : u = u \...
Lucas Linhares's user avatar
2 votes
0 answers
45 views

Asymptotic behavior of the solution of homogeneous heat equation with Neumann boundary condition.

For $$ \begin{cases}u_t=\Delta u & \text { in } \Omega \times(0, \infty), \\ \frac{\partial u}{\partial \nu}=0 & \text { on } \partial \Omega \times(0, \infty), \\ u(x, 0)=\phi(x), & x \in ...
Elio Li's user avatar
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1 answer
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Is it possible to find an ellipse equation while given 3 arbitrary points on it? (no center or foci are given)

Hi I am dealing with an image processing issue where I have parts of an ellipse visible (which is a circle projection) while other parts of an ellipse are blocked by another objects on a scene. I ...
Igor's user avatar
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0 answers
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Concluding a radial weak solution with radial test functions is a weak solution with all test functions

Let $B \subset \mathbb{R}^N$ be the unit ball and $E = \{u \in H^1_0(B) : u \text{ is radial}\}$. Define the functional $I : E \to \mathbb{R}$ by $$ I(u) = \int_{B} |\nabla u(x)|^2 dx - \frac{1}{p}\...
Lucas Linhares's user avatar
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0 answers
16 views

Solvability of the oblique derivative boundary value problem

Let $\Omega$ be a bounded $C^{2,\alpha}$ domain in $\mathbb{R}^n$ that satisfies an interior sphere condition at each point of $\partial \Omega$, i.e., for each $x_0\in \partial \Omega$, there exists ...
Stephen_lamb's user avatar
1 vote
0 answers
48 views

Solving elliptic equations on the whole space?

I am interested in showing the existence of solutions to elliptic equations of the form $$-\Delta u(x) + q(x) u(x) = f(x)$$ with $q \in L^\infty(\mathbb{R}^n)$ (perhaps even $C^\alpha(\mathbb{R}^n)$) ...
Dominic Shillingford's user avatar
1 vote
1 answer
130 views

Regularity of weak solution of elliptic equation with nonlinear Neumann boundary

Let $\Omega \subset \mathbb{R}^n$ be bounded smooth domain and $u \in W^{1,2}(\Omega;\mathbb{R}^m)$ be a weak solution to the following equation \begin{align*} &\int_{\Omega} \nabla u \cdot \nabla ...
mnmn1993's user avatar
  • 395
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0 answers
35 views

Difference of boundary conditions between elliptic and parabolic PDEs

For elliptic PDEs we know that the boundary condition no matter Dirichlet, Neumann or mixed, needs to be defined over the whole boundary of the given volume $\partial V$. Only then we will get a ...
Krum Kutsarov's user avatar
1 vote
1 answer
36 views

Converting the ellipsoid from quadratic matrix form to a parametric form to find x, y and z co-ordinates

I have an ellipsoid as given here: $(p-C)^T Q(p-C)=1$ The terms $p, C$ and $Q$ are defined in the linked page. My goal is to find equations for the $x,y,z$ co-ordinates of the resultant ellipsoid in a ...
wanderer's user avatar
  • 105
2 votes
1 answer
46 views

A property of solution operator of a elliptic PDE involving positive part of a function

For a given $u \in L^2(\mathbb{R}^N)$, there is a unique $S(u) \in H^1(\mathbb{R}^N)$ such that $$ \int_{\mathbb{R}^N} \nabla S(u) \nabla \varphi + S(u) \varphi = \int_{\mathbb{R}^N} u \varphi, \quad \...
Lucas Linhares's user avatar
1 vote
0 answers
22 views

Positive part of the solution - relation between $B(u)$ and $B(u^+)$

Given $u \in L^2(\mathbb{R}^N)$, by Riesz Representation Theorem there exists a unique $B(u) \in H^1(\mathbb{R}^N)$ weak solution of $$ -\Delta v + v = u, \quad \mathbb{R}^N. $$ Given $u \in L^2(\...
Lucas Linhares's user avatar
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0 answers
36 views

Restriction of solution operator of a elliptical equation in $\mathbb{R}^N$

For a given $u \in L^2(\mathbb{R}^N)$, we know by Riez Representation Theorem that there exists a unique $B(u) \in H^1(\mathbb{R}^N)$ such that $$ \int_{\mathbb{R}^N} \nabla B(u) \nabla \varphi + B(u) ...
Lucas Linhares's user avatar
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31 views

Positiveness of the solution of a system of elliptical equations

Given $u \in L^2(\mathbb{R}^N)$, by Riesz Representation Theorem we know there exists a unique $v=B(u)\in H^1(\mathbb{R}^N)$ weak solution of $$ (1) \,\,-\Delta v + v = u, \mathbb{R}^N. $$ Suppose ...
Lucas Linhares's user avatar
2 votes
3 answers
90 views

Generalization of $\max$ and $\min$ of $Ax^2+Bxy+Cy^2$ with given $x^2+y^2=k$.

Generalization of $\max / \min (Ax^2+Bxy+Cy^2)$ with given $x^2+y^2=k$. ($A, C > 0$) My first thought was drawing an ellipse and circle of each of the 2 given equations and think about the ...
RDK's user avatar
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0 votes
1 answer
51 views

find an ellipse equation satisfied

In the standard coordinate system of $E^2$, given the second-degree curve S with the equation: $$5x^2+8xy+5y^2-18x-18y+9=0$$ a) Prove that $(S)$ is an ellipse. b) Write the equation of the ellipse ...
lee max's user avatar
  • 249
0 votes
0 answers
25 views

Sobolev inequality in the caccioppoli inequality for elliptic systems with superquadratic controllable growth conditions

The line in the proof that confuses me Hey guys, I stumbled upon this line in the proof for the caccioppoli inequality for elliptic systems with superquadratic controllable growth conditions. The ...
MathsUser1's user avatar
6 votes
1 answer
177 views

How does one convert the mean curvature equation into a homogeneous linear elliptic P.D.E?

In Henry C. Wente's paper https://msp.org/pjm/1980/88-2/pjm-v88-n2-p08-s.pdf, page $391$ to $392$, the following prescribed mean curvature equation is considered: $$\mathrm{div}(Tu)=nH(x,u),\qquad \...
Tiffany's user avatar
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1 vote
0 answers
47 views

Confusion on definition of fundamental solution for the heat equation

As mentioned on Wikipedia, a fundamental solution for a linear differential operator $L$ is a function (or distribution) $G$ such that $$LG = \delta$$ which by linearity of $L$ gives the following ...
CBBAM's user avatar
  • 6,265
0 votes
0 answers
29 views

On Schauder estimates and harmonic coordinates in Petersen's book

In Peter Petersen's Riemannian Geometry(3rd ed), Section 11.2, following form of local Schauder estimate is stated without proof (Theorem 11.2.2): For elliptic 2nd order differential operator $L$ on ...
Kiyoon Eum's user avatar
0 votes
0 answers
7 views

A inequality involving an exponential non linearity of a PDE

Let $f : \mathbb{R} \to \mathbb{R}$ a continuous function and suppose there exists $\alpha_0 > 0$ such that $$ \lim_{|s| \to +\infty} \frac{|f(s)|}{e^{\alpha |s|^{N/(N-1)}}} = 0, \quad \forall \...
Lucas Linhares's user avatar
0 votes
1 answer
71 views

How to calculate the Volume of an Elliptic Truncated Cone?

I have been attempting to find the volume of an Elliptic truncated cone by dividing it into cross-sections of elliptical cylinders and then stacking them up. I got the idea from the integration of the ...
Artimieus's user avatar
2 votes
0 answers
54 views

Second order elliptic PDE with inhomogeneous Dirichlet boundary condition

For second order elliptic PDE, we usually consider the weak solution, which is the solution for variational form. When facing inhomogeneous Dirichlet boundary conditions, we usually modify it into ...
GeneLIU's user avatar
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4 votes
0 answers
90 views

The constant in Schauder estimate of linear elliptic PDE

Assume that $u(x)$ is the classical solution solving $$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$ on $\mathbb{R}^n$ for some smooth enough coefficients and uniformly elliptic $a_{...
mnmn1993's user avatar
  • 395
4 votes
1 answer
57 views

Is it possible to find a function $u_0 \in H_0^{1}(\Omega)$ such that $I(u_0)<0$?

Let $\Omega \subset \mathbb{R}^n$ be an bounded domain with smooth boundary $\partial \Omega$ and consider the functional $I:H_0^{1}(\Omega) \to \mathbb{R}$ defined by $$I(u)=\frac{1}{2}\lVert u \...
Victor's user avatar
  • 289
0 votes
0 answers
33 views

Dense subspace of radial functions of $H^1_0(\Omega)$

Let $\Omega\subset \mathbb{R}^N$ a bounded domain. By definition the space of radial functions in $H^1_0(\Omega)$ is $$ H^1_{0, rad}(\Omega) = \{u \in H^1_0(\Omega) : u = u \circ R, \forall R \in O(N)\...
Lucas Linhares's user avatar
0 votes
0 answers
14 views

A question about regularity results in the linear Elliptic case which are given by Schauder theory

I've been reading Jost's lecture notes "Nonlinear Methods in Riemannian and Kählerian Geometry". In section 2.2 he gives a regular results about Elliptic and parabolic equations, but he ...
luyao's user avatar
  • 13
0 votes
1 answer
68 views

$-\Delta u =1$ in a unbounded strip

Consider the problem $$ \left\lbrace \begin{gathered} -\Delta u = 1 \quad\textrm{in}\quad S\\ u=0\quad\textrm{on}\quad \partial S \end{gathered} \right. $$ where $S$ is the infinity strip $\mathbb{R}\...
user143090's user avatar
1 vote
0 answers
47 views

Linear elliptic problem inverse mapping is Lipschitz in log permeability

I am reading this paper and would like to check how they derive the inequality in (5) on page 4. Denotes $S^d$ the set of symmetric second order tensors on $\mathbb{R}^d$. Define the permeability ...
Q-Y's user avatar
  • 1,589
1 vote
2 answers
52 views

Global $L^{2}$ solution of Laplace equation in $\mathbb{R}^{3}$.

Does there exists a global and non-trivial solution $u\in L^{2}(\mathbb{R}^{3})$ to the Laplace equation $\Delta u=0$? Using the spherical symmetry of the problem, one can consider the usual solution ...
B.Hueber's user avatar
  • 2,876
2 votes
1 answer
72 views

FEM for non linear PDEs

I am looking for an easy but rigorous reference for FEM methods for non linear PDEs like the p-laplace equattion or non linear heat equation ect. Can one recommend me a good exposition of this topic ...
Furkan's user avatar
  • 69
0 votes
0 answers
41 views

Maximum principle to get $C^0$ estimates in terms of $L^1$ norms

I'm reading this paper https://arxiv.org/abs/1401.7366 and trying to prove Corollary 4.6. The result essentially says the following. Let $B_r \subseteq \mathbb R^4$ be the ball of radius $r$ with the ...
Holmes's user avatar
  • 906
1 vote
1 answer
65 views

How we get the jacobi elliptic functions [closed]

I am encountering an issue with Jacobi elliptic functions. Specifically, I am facing a challenge with the following integral: $$\pm\int_{-\sqrt{\frac{2\mu}{\eta}}}^{u}\frac{\mathrm ds}{s\sqrt{\left(s+\...
Espoir's user avatar
  • 15
0 votes
1 answer
73 views

Type of solution (as the critical point of the energy functional) after doing scaling to the PDE.

Now I have a nonlinear elliptic PDE, if I do scaling such as $u=\lambda v$ and then get a new PDE, then I turn to study the new equation since it has a simpler form. My question is easy to understand: ...
Elio Li's user avatar
  • 617
0 votes
0 answers
19 views

Bound by limit plane of harmonic function on half space vanishing at infinity

The problem is Suppose $\phi\in\mathcal{S}(\mathbb{R}^2)$, $u$ on $\mathbb{R}^3_-=\mathbb{R}^2\times(-\infty,0)$ satisfies \begin{cases} \Delta u(x,y,z)=0\\ \lim\limits_{z\to 0^-} u(x,y,z)=\phi(x,y)\\ ...
vegetabledoge's user avatar
3 votes
0 answers
196 views

Gradient estimates of linear elliptic PDE

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain. Assume that $u(x)$ is the classical solution solving $$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$ $$u(x)\Big|_{\...
mnmn1993's user avatar
  • 395
2 votes
0 answers
170 views

$W^{1,p}(\Omega)$ estimates of solutions of elliptic equation

Consider the following simple elliptic problem in a bounded domain $\Omega\subset\mathbb{R}^N$: $$ -\Delta u(x) + u(x) = f(x) \qquad \forall x\in \Omega $$ with Neumann boundary conditions in $\...
joaquindt's user avatar
0 votes
1 answer
38 views

Mixed Laplace equation and missing one step in the integration by parts

I am looking at the Deal.ii finite element package docs, and was reading the section about vector valued problems. These are simply systems of PDEs that are solved together. I am trying to derive the ...
krishnab's user avatar
  • 2,531
3 votes
1 answer
70 views

Construct a sequence of real analytic functions solving the equation with a condition

Consider the Laplace equation in two dimensions: $$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0$$. Construct a sequence of real analytic functions $(u_k)_{k=1}^\infty$ with $u_k:...
Bagaringa's user avatar
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