Questions tagged [elliptic-equations]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
12 views

Calderon-Zygmund in solving $\Delta u + (d f, \theta)_{\omega}=e^{u}$ when we have the $L_{\infty}$ estimation of $u$ on complex compact manifold

I'm reading the paper On Chern-Yamabe problem in this paper, $\Delta^{C h} f=\Delta_{d} f+(d f, \theta)_{\omega}$, after getting the uniform $L_{\infty}$ bound of the constructed sequence $f_{t_{n}}$, ...
user avatar
0 votes
0 answers
25 views

Calculate an ellipsoid given two points and a arc length.

I have a problem and I would like some insight into if people think it is possible. If they think it is possible do they also have an idea of how to solve it. Given two points $A$, $B$ in the ...
user avatar
  • 13
0 votes
0 answers
31 views

Question about higher order mean field equation $\left(-\Delta_{g}\right)^{m} u+\lambda=\lambda \frac{e^{2 m u}}{\int_{M} e^{2 m u} d \mu_{g}}$ .

I'm reading Dr.Luca Martinazzi's paper Existence of solutions to a higher dimensional mean-field equation on manifolds which proves that for $m \geq 1$, there is an existence result for the equation $$...
user avatar
0 votes
1 answer
32 views

Estimate the norm of the solution of an elliptic partial differential equations of second order

Let $\Omega$ be a bounded domain of $\mathbb{R}^n$, $\partial \Omega$ is of class $C^{2}$ . Assume that $a_{ij}(x)\in C^{0,1} (\bar{\Omega}), \Lambda|\xi|^2> a_{i j}(x) \xi_{i} \xi_{j} >\...
user avatar
  • 21
1 vote
0 answers
39 views

Problems about moving plane method

Recently I am reading Elliptic Partial Differential Equations by Qing Han and Fanghua Lin, in page 44, the authors proved the moving plane method, I have got some ideas about its generalization but I ...
user avatar
  • 21
1 vote
0 answers
24 views

Linear Elliptic PDE Variable Coefficients Non-Separating Variables

I am trying to obtain Analytical solutions for the following Linear Elliptic PDE in the dependent variable U(x,y) having variable coefficients. 'x' is a (pseudo) radial coordinate, and 'y' is an ...
user avatar
3 votes
0 answers
76 views

Fundamental solution of Helmholtz equation in $L^p$? why?

Let $N \geq 2$ an natural number and let $k>0$. Consider de equation defined on $\mathbb{R}^N\setminus \{0\}:$ $$\Delta u(x)-k^2u(x)=0.$$ It is known that if $k=1$ then $E_N(x)=\gamma_N |x|^{1-N/2}...
user avatar
0 votes
0 answers
17 views

How many positive harmonic functions are there on a half-space for a vanishing boundary condition

Given a space $\mathbb{R}^n$, can I say all functions $$u: \{x_1\geq 0\}\to \mathbb{R}$$ s.t. u is harmonic on $\{x_1\geq 0\}$, u is non-negative and u vanishes on the boundary $\{x_1=0\}$ is exactly $...
user avatar
1 vote
0 answers
32 views

Elliptic regularity with right-hand side in $H^{-1/2}$

If $\mathbf{f} \in H^{1/2}(\Omega)$ for a bounded domain $\Omega$ with smooth boundary, does the elliptic regularity for the Laplacian guarantee that the solution to the elliptic problem in divergence ...
user avatar
  • 88
0 votes
0 answers
24 views

Finite differences and Finite Element Method

I have the following 1D problem: \begin{equation} \begin{cases} -u''=f \ \ \ x\in(0,1)\\ u(0)=u(1)=0 \end{cases} \end{equation} I have derived the Galerkin formulation and I have implemented a code ...
user avatar
  • 447
0 votes
0 answers
20 views

Liouville's equation $\Delta u=K e^{ u}$ when $K<0$.

I'm quite interested in the elliptic PDE like $\Delta u = K e^{u}$ when $K<0$, when $K>0$, it's very easy talk about the existence of the solution but when $K<0$ it seems that we lose some ...
user avatar
0 votes
0 answers
6 views

Covering argument for linear elliptic equations

I am looking for some ideas on trying to show if the following sort of result is true or false: Let $u$ be a bounded, local, weak solution of $\text{div} A(x) \nabla u = 0$. Since $u \in W_{loc}^{1,2}...
user avatar
  • 257
0 votes
0 answers
25 views

How to classify this PDE? (similar to free boundary problem)

For some bounded smooth domain $D$, I woud like to solve the problem $$ \Delta u = f(x) \quad \text{ in \{ $x \in D$ :$u>0$\}}$$ with $u\Big|_{\partial D} = \phi(x)$. Here $f(x)$ and $\phi(x)$ are ...
user avatar
  • 533
1 vote
0 answers
19 views

Elliptic regularity on the cube

Does anyone know if elliptic regularity still holds in the d-dimensional cube $Q:=[0,1]^d$ with Neumann boundary conditions ? More precisely, let $\rho\in C^{0,\alpha}(Q)$ (let say scalar) with $\...
user avatar
  • 21
0 votes
0 answers
22 views

Regularity wrt. time of time dependent elliptic pde

lets say I have the following problem: $\Delta u(x,t) = f(x,t), x\in\Omega\\ u(x,t) = g(x), x\in \partial \Omega, t\in[0,T]$ and $f\in C^\infty([0,T];L^2(\Omega))$ and $\Omega$ is a Lipschitz domain ...
user avatar
  • 1
0 votes
0 answers
15 views

Equivalence in definition of ellipticity for fully nonlinear pde operators

I'm reading Ros-Oton's book on regularity for elliptic PDE, and when in the context of a fully nonlinear operator, one studies the following $$ F(D^2u) = 0 $$ and defines $F$ to be uniformly elliptic ...
user avatar
0 votes
0 answers
43 views

About a remark in Evans PDE, in the regularity of elliptic equation

On Page 341, after Theorem 5 (Higher boundary regularity) in Chapter 6.3 (Regularity), Evans states that Remark. If $u$ is the unique solution of $$ \left\{\begin{aligned} L u=f & \text { in } U \...
user avatar
  • 23
0 votes
0 answers
34 views

On the elliptic regularity theory of a Schrödinger equation

Let $u\not\equiv0$ be a non-negative solution of the following equation: $$ -\Delta u+u=|u|^{p-2}u,\quad u\in H^1(\mathbb{R}^N), $$ where $N\geq3$ and $2<p<2^*=2N/(N-2)$. My goal is to prove ...
user avatar
  • 319
0 votes
0 answers
20 views

Choose Green's function of Possion equation on half-space

Background: Consider the Dirichlet problem (A) defined in $\mathbb{R}^3$ $$-\Delta u=0~~~~\text{in $\mathbb{R}_+^3$}=\{x=(x_1,x_2,x_3):x_3>0\},$$ $$u=g~~~~\text{in $\partial \mathbb{R}_+^3$ }.$$ We ...
user avatar
  • 203
2 votes
1 answer
54 views

Show that $\int_0^{\pi/2} \frac{dt}{\sqrt{\sin t}}=\int_0^1 \frac{dx}{\sqrt{x-x^3}}$

This is intended to be an example of an elliptic integral but I'm not sure how to go about showing it. I'm not sure which identities to start with, any tips would be greatly appreciated!
user avatar
0 votes
0 answers
53 views

A sequence of solutions of elliptic equations converges uniformly on compact subsets

$\textbf{Theorem 3.}$ $\Omega$ is a bounded domain in $\mathbb{R^2}$. Suppose $(u_n)$ is a sequence of solution of $$-\Delta u_n=V_n(x)e^{u_n}\quad\text{in}\quad \Omega $$ for some $1<p\leq \infty$....
user avatar
0 votes
0 answers
31 views

Existence of Weak Solution of the Neumann Problem with non zero boundary data

I need to show that there is a unique solution modulo constants to the following Neumann problem: $\Delta u=0$ in $\Omega$ and $\frac{\partial u}{\partial \nu}=h\in H^{-1/2}(\partial \Omega)$ where $\...
user avatar
  • 73
2 votes
0 answers
33 views

What are the existing results on the $C^{0}$ estimate of the solution of the semi-linear pde $\Delta u = f(x,u,\nabla u)$.

I have read several papers but their conditions are so strong which always assume that $f(x,z,\eta)$ is controlled by $\left| \eta \right|$, for example: $\left| f\right| \leq C(\left| \eta \right|^{2}...
user avatar
0 votes
1 answer
22 views

Quotient of solutions of a semilinear Dirichlet problem is $L^\infty$

I am reading a paper of Brezis and Oswald about existence and uniqueness of positive solutions to sublinear elliptic equations: \begin{equation} - \Delta u = f(x, u), \ u \geq 0, \ u \not\equiv 0 \...
user avatar
0 votes
0 answers
24 views

Constant solution for an elliptic equation

Let $U$ be a open and connected subset of $\mathbb{R}^n$ with regular boundary. Consider the fallowing elliptic problem $$ \Delta u + c(x) u = u^3, U\\ \hspace{2cm}u = 0, \partial U, $$ where $c$ may ...
user avatar
  • 301
1 vote
1 answer
64 views

Reference for $L^p$ estimates

My PDE professor showed the following result: Let $f \in L^{p}(\Omega)$, for $1 < p < \infty$. Also consider $u \in L^{1}_{loc}(\Omega)$ a solution of \begin{align} -\Delta u + a(x)u&= f, \...
user avatar
  • 301
0 votes
0 answers
29 views

Estimate the error caused by domain perturbation for Dirichlet Laplacian equation

$\Omega$ is a bounded smooth domain in $\mathbb{R}^N$, consider the Dirichlet laplacian equation on $\Omega$. \begin{cases} -\Delta u=f & \mbox{in }\Omega\\ u|_{\partial \Omega}=0 \end{cases} ...
user avatar
  • 322
0 votes
0 answers
20 views

Question about the use of the Min-Max method in solving evolution equations.

in elleptic PDE's i used Min-Max methods and critical points researche to find my weak equations and i learnd about their techenics however, i didn't find up to this time references speaking of this ...
user avatar
2 votes
0 answers
43 views

How to prove the following superquadratic property

Let $\Omega \subset \mathbb{R}^N$ a bounded smooth domain and let $f:\Omega \times \mathbb{R} \to \mathbb{R}$ an Carathéodory function such that $(f_1): |f(x,s)| \leq c|s|^{\sigma} + d$ for all $x \in ...
user avatar
0 votes
0 answers
22 views

A variational argument involving the Rayleigh quotient from Gilbarg-Trudinger

I am currently reading Gilbarg-Trudinger's discussion of the eigenvalues of a self-adjoint scond-order elliptic operator. Let $Lu = D_i(a^{ij}D_ju + b^iu) - b^iD_iu + cu$ with $a^{ij}$ symmetric. ...
user avatar
  • 189
2 votes
1 answer
62 views

An interior estimate for the gradient of weak solutions to elliptic equations

Let $L$ be an elliptic operator in divergence form i.e. $$Lu = -\sum_{i,j=1}^n D_i(a^{ij}(x)D_ju) + \sum_{i=1}^nb^i(x)D_iu + c(x)u$$ with $a^{ij}, b^i, c \in C^1(Ω) ∩ L^ ∞(Ω)$ and $L$ uniformly ...
user avatar
  • 189
1 vote
1 answer
52 views

Why do weak $H^2_{\mathrm{loc}}$ solutions to elliptic equations solve the PDE almost everywhere?

I am reading Evans' Partial Differential Equations and encountered something I don't understand: Let $u \in H^1(U)$ be a weak solution to a second order elliptic PDE with nice enough coefficients $Lu =...
user avatar
  • 189
0 votes
1 answer
31 views

On the existence of global classical non-zero solutions of a linear elliptic equation

Does the equation $$-\Delta u +u=0$$ have any non-zero classical, i.e., $C^2$, solutions on $\mathbb{R}^d$? How about if $\mathbb{R}^d$ is replaced with half-space? How about solutions of polynomial ...
user avatar
  • 245
0 votes
1 answer
23 views

Angle of line at known point on known ellipse

I have an oval/ellipse, $x^2/a^2 + y^2/b^2 = 1$. The ellipse is always centered at $(0,0)$. I have a point on that ellipse, $(x, y)$. I'm looking for the angle of the tangent at that point relative to ...
user avatar
1 vote
1 answer
42 views

Harmonic Function and Subharmonic function

Suppose $u\in C^1(\bar{U})\cap C^3(U)$, where $U$ is a bounded simply connected open set. If $\Delta u=0$ and $u(x)\neq0$ for all $x$ , show that $\varphi=\frac{\vert Du\vert^2}{u^{\frac{2(n-1)}{n-2}}}...
user avatar
  • 61
0 votes
1 answer
33 views

An application of mean value theorem in Banach spaces

I am reading this paper and I did not understand a statement done in the final of the page $581$: $$I_{Q_{\lambda}}(z_{\delta}) - I_{Q_{\lambda}}(u_n) \geq - \frac{1}{n} ||z_{\delta} - u_n||$$ and by ...
user avatar
  • 3,437
1 vote
0 answers
118 views

Estimating the $L^{\infty}$-norm of a Laplacian of a function

Considering the problem \begin{equation} \left\{ \begin{array}[c]{11} \Delta(\Delta \chi -\chi) = 0 & \text{in } \Omega, \\ \Delta \chi -\chi = h_2 - h_1, & \text{on } \partial\Omega \\ \end{...
user avatar
  • 443
3 votes
1 answer
118 views

Solve double integral $\iint_E\frac{1 + x}{1 + 2x^2 + 3y^2} dA$ ellipstic area $E$

I am trying to solve the integral $$\iint_E\frac{1 + x}{1 + 2x^2 + 3y^2} dA$$ where E is the elliptical area given by $$2x^2 + 3y^2\leq 6$$ I have tried substitution with $(x, y)\mapsto(\sqrt{3} r\cos\...
user avatar
5 votes
1 answer
109 views

temperature distribution using Fourier/Laplace

Consider room that occupies quarter space That is formed by two walls. One wall is fully insulated and has a constant temperature $0$. Another wall has a window of length L that with one edge at the ...
user avatar
0 votes
2 answers
51 views

What are the equations of rotated and shifted ellipse, parabola and hyperbola in the general conic sections form?

How will look the general conic sections equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ in case of rotated and shifted from coordinates origin ellipse, parabola and hyperbola? I need a formulas for ...
user avatar
  • 189
1 vote
0 answers
145 views

How to approach this semilinear system of PDEs?

Cross-posted to MathOverflow (link) My question concerns a system of $n$ $n$-dimensional PDEs. \begin{equation*} -\frac{1}{2}\sum_{jk}a_{jk}x_{j}x_{k}u^{i}_{x_{j}x_{k}} +\sum_{jk} u^{j}\pi_{jk}u^...
user avatar
  • 714
1 vote
1 answer
81 views

Elliptic PDE existence theorem for $f \in H^{-1}(\Omega)$

Let $\Omega \subset \mathbb{R}^N$ be a bounded open set and consider the elliptic PDE $$ \begin{align} Lu = f \quad &\textrm{in } \Omega, \\ u = 0 \quad &\textrm{on } \partial \Omega, \end{...
user avatar
  • 111
0 votes
0 answers
19 views

Estimate for Green's function in $\Omega \subset \mathbb{R}^2$. Can we show that $|\nabla G(x,y)| \le \frac{C}{|x-y|} $?

Assume $\Omega$ is a smooth bouned domain of $\mathbb{R}^2$. Let $$G(x,y)=\frac{1}{2\pi} \log{\frac{1}{|x-y|}}+H(x,y)$$ be the standard Green's function on $\Omega$. I want to show that $$|\nabla H(x,...
user avatar
  • 123
0 votes
0 answers
51 views

Solving Laplace's Equation on Complex Geometries Using Conformal Maps

I am interested in solving Laplace's equation $$\Delta u = 0.$$ with Dirichlet boundary conditions but on complex geometries. To my understanding, the standard approach is to (if possible) map your ...
user avatar
  • 1,366
1 vote
1 answer
59 views

Linearisation of a Quasilinear Elliptic PDE

I have noticed that the word 'linearisation' can have different meanings in different places in the literature. For example, if one has a second-order quasilinear elliptic PDE, what would be the ...
user avatar
  • 1,076
1 vote
0 answers
49 views

Apparent error in Jost's Partial Differential Equations

Theorem 13.1.1 in Jost's Partial Differential Equations asserts that if $f \in L^\infty(\Omega)$, with $\Omega$ a bounded open set in $\mathbb{R}^2$, then $$ u(x) = \int_\Omega \log |x-y| f(y)\ dy $$ ...
user avatar
0 votes
1 answer
19 views

Can we find a uniform bound of the solution of a series of linear partial differential equations related to a parameter.

Let $\sigma \in[0,1]$,we consider following series of linear partial differential equations related to the parameter $\sigma$,for example $$ \left\{\begin{aligned} \Delta \Phi &=\sigma f(x, y) \...
user avatar
1 vote
0 answers
35 views

Average behavior of Green's function near the boundary

Let $\Omega \subset \mathbb R^d$ be a Lipschitz domain. Let $g$ be the Green function of $\Omega$ for the operator $\operatorname{div}(A\nabla \cdot)$ ($A$ with $C^\infty$ coefficients or, for ...
user avatar
  • 251
2 votes
0 answers
35 views

Variable transformations in subsolutions of elliptic equations

I am reading Qing Han,& Fanghua Lin’s elliptic partial differential equations. The book had the following variable transformations. (proof of theorem 4.1) Suppose that $u\in H^1_0(B_1)$ is a ...
user avatar
  • 67
1 vote
1 answer
84 views

Uniformly elliptic implies existence of unique weak solution

Provide a weak formulation for the following boundary value problem $$ (*) \quad L u=: \sum_{i, j=1}^{n} \partial_{j}\left(a_{i j} \partial_{i} u\right)+c u=f, \quad \text { in } \Omega, \quad \text { ...
user avatar

1
2 3 4 5
11