Questions tagged [regularity-theory-of-pdes]
This tag is for questions concerning the smoothness of weak solutions to partial differential equations.
746
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Regularity for heat equation with Neumann boundary conditions
Consider the following inhomogeneous heat equation for $f=f(x,t)$ with Neumann boundary conditions on a bounded domain $\Omega$ and initial value $f(x,0)=f_0(x)$:
\begin{align*}
\partial_t f - \Delta ...
- 11
0
votes
0
answers
22
views
De Giorgi-Nash-Moser estimattes for non-symmetric a_ij
Is there any references on De Giorgi-Nash-Moser estimates for the parabolic operator
$$ \partial_t - \partial_j[a_{ij}(t, x)\partial_i] $$
where $a_{ij}$ is not symmetric?
- 11
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0
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29
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Bound for $\left\|\nabla^2 u\right\|_{L^2(\Omega)}$
Consider the elliptic equation
$$
\begin{aligned}
-\nabla \cdot A \nabla u=f, & \text { in } \Omega, \\
u=0, & \text { on } \partial \Omega.
\end{aligned}
$$
where $\Omega$ is a bounded domain....
- 435
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0
answers
79
views
Heat equation with sharp inhomogenous Dirichlet conditions
Consider the standard heat equation in with Dirichlet boundary data.
$$\begin{cases}
\partial_tu - \Delta u = 0 & \text{in } \Omega_T := (0,T) \times \Omega, \\
\hfill u = g & \text{in } ...
- 6,997
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2
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43
views
Regularity of solutions of one kind of elliptic PDE
Consider $\Omega$ a bounded domain with $C^{1}$ boundary and the elliptic PDE
$$
-div (F(\nabla u)) = \lambda|u|^{p-2}u,\ u \in W^{1,p}_0 (\Omega).
$$
What kind of conditions on $F$ one must have in ...
2
votes
0
answers
53
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On the $L^3$ norm for the Navier Stokes equations
I'm reading the paper of Seregin, Euscariaza and Sverak about the smoothness of weak solutions $(u,p)$ of the Navier Stokes equations when $u \in L^{\infty}([0,T],L^3(\mathbb{R}))$.
Here is the link.
...
- 83
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33
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Maximum principle for heat equation in Brezis' Book
In Brezis' book, look at the maximum principle for the heat equation in $\Omega \times (0,T)$ where $\Omega$ is an open bounded subset of $\mathbb{R}^{n}$:
$$u_{t}-\Delta u=0$$
$$ (x,t) \in \Omega \...
- 31
0
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0
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14
views
Heat semigroup in Heisenberg groups properties
I'm trying to find the construction of the heat semigroup in Heisenberg groups, but I haven't found anything. In particular, I'm trying to find out if the semigroup in the Heisenberg group has the ...
- 2,817
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0
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10
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regularity of solution in banach valued space
hi im studying heat equation in open,bounded,regular $\Omega \subset R^{n}$
$$\partial_{t}u-\Delta u=0 \ (x,t)\in \Omega \times (0,\infty)$$
with
$$u=0 \in \partial \Omega\times (0,\infty) and \ u_{...
- 31
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0
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question about Claude Zuily book chapter heat eqaution
i found in claude Zuily's book chapter heat equation (french book) that for$\Omega$ bounded regular ,open $\subset R^{n}$ if we consider the homogenous heat equation given by:$$\partial_{t}u-\Delta u=...
- 31
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0
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15
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banach space and sobolev valued space
i know that $ L^2 (0, T; H^1_0(\Omega))$ and $C (0, T; H^1_0(\Omega))$ when T is finite are banach spaces can we say the same thing if $T =\infty$?
i read also in evans book that $ L^{2}(0, T; H_{...
- 31
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Reference request: existence, uniqueness, and regularity of solutions to elliptic PDEs with periodic boundary conditions
I am interested in Theorems regarding existence, uniqueness, and regularity of solutions to linear 2nd order elliptic PDEs with periodic boundary conditions, e.g.,
$$\begin{cases} -u''+u=f & 0<...
- 22.3k
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0
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30
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uniqueness for non homogenous heat equation
hi in nonhomogenous heat equation :$u_{t}-\Delta u=f(x,t)$ in U smooth open of $R^{n}$ and $f \in L^{2}((0,T),L^{2}(U))$ with $u_{0}$ in $L^{2}(U)$,$u=0$ in $\partial U$
i find that u $\in C((0,T),L^...
- 11
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question about proof of theorem of regularity
i found a theorem saying that if u belongs to $H^{1}(\Omega)$ where $\Omega$ is bounded regular domain of $R^{n}$ then $\nabla u^{+}=\chi_{u>0}\nabla u $ here $u^{+} =sup(0,u)$
the hint is to use ...
- 11
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Does the regularity theorem for linear parabolic equations apply for any compact smooth manifold as well?
In Evans' PDE bok (Second Edition) p.384 Theorem 5, it is stated essentially that the weak solution $u(x,t) : U \times [0,T] \to \mathbb{R}^n$ of the parabolic equation
\begin{equation}
\partial_t u -\...
- 6,808
0
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0
answers
48
views
Solving for pressure in NS equation
Let us consider the Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces provided by the formula:
\begin{align}
\dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\...
- 59
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0
answers
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Existence and regularity for an elliptic equations
I am reading a paper in which the authors recall (without proving) the existence and regularity of solutions to this equations
$$-\Delta y + y +\nabla p = f \quad \text{ in } \Omega$$
$$\mathrm{div}\, ...
- 33
0
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1
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How is the viscosity solution is $C^2$ away from $z=0$. The domain $B_R \cup {z \neq 0}$ already contains the point $z=0$.
In Caffarelli and Silvestre's extension problem paper link of paper Lemma 4.2, in Lemma 4.2, it is given that the viscosity solution is $C^2$ away from $z=0$. How do we have a classical solution on $...
1
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31
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Parabolic regularity on a $C^\infty$ compact manifold
I have a general parabolic PDE on a compact $C^\infty$ manifold $M$ $\textit{ie}$ a (the?) solution $u$ is a probability density
and satisfies the PDE in a weak sense (with $C^\infty_c$ test functions)...
- 41
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29
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Proving that a $W^{1,2}$ type space is Hilbert
Split the $N-$dimensional Euclidian space as $\mathbb{R}^N = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$. A vector in $\mathbb{R}^N$ will be denoted by $z = (x,y)$. Let $\alpha > 0$ and consider the ...
- 843
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33
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Caffarelli-Silvestre Extension problem (Existence of Solution of Degenerate elliptic PDEs in a ball)
In the Caffarelli-Silvestre paper link of paper before Lemma 4.2, it is given that \begin{align} \text{div}(|y|^a \nabla u)=0 ~~~\text{in the weak sense in} ~B_R\\ u=g \hspace{3.3cm}\text{on}~\partial ...
1
vote
0
answers
14
views
About $\epsilon$-Regularity theorem in Leon Simon's Theorems on Regularity and Singularity of Energy Minimizing Maps
I am reading Theorems on Regularity and Singularity of Energy Minimizing Maps by Leon Simon. The problem arises in Section 2.3, the proof of Shoen-Uhlenbeck theorem:
Let $u:\Omega \to N$ is a ...
- 33
2
votes
1
answer
41
views
References for regularity of solution to fourth order elliptic/parabolic PDEs
After searching, all I could find was regularity of solutions to second-order PDEs like in Partial Differential Equations by Evans and Functional Analysis, Sobolev Spaces and Partial Differential ...
- 75
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votes
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34
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A continuous function in $W^{1,p}(I)$ function with continuous weak derivative is a $C^1(\overline{I})$ function
In Theorem 8.2. of Brezis's book of Functional Analysis, says that a function $u \in W^{1,p}(I)$ has a continuous representante $\tilde{u}$ such that
$$
\int_y^x u'(t) dt = \tilde{u}(x) - \tilde{u}(y)....
- 843
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40
views
Matching conditions of locally $H^2$ functions to ensure global $H^2$ regularity
Let $\Omega_1$ and $\Omega_2$ be a non-overlapping partition of a domain $\Omega$. Suppose that a function $u \in L^2(\Omega)$ has $H^2$ regularity on the two subdomains (i.e. $u|_{\Omega_1} \in H^2(\...
2
votes
1
answer
66
views
If $f$ is a eigenfunction of $-\Delta$ in $L^2[0,1]$, is it necessarily $C^\infty$?
I am a little bit confused about the properties of the Laplacian $-\Delta$ on $L^2[0,1]$ with the periodic boundary conditions.
At least I know that $-\Delta$ is an unbounded self-adjoint operator on $...
- 6,808
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1
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Regularity of solution of ODE $\frac{d}{dx}\left(\frac{1}{n^2}v'\right)+k^2 v = 0$
Given ODE: $$ \frac{d}{dx}\left(\frac{1}{n^2}v'\right)+k^2 v = 0, x \in [a,b] $$ where $n$ is a discontinuous function in $[a,b]$ such that $n^2$ is discontinuous and non-zero (take a step function ...
- 1,673
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votes
1
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32
views
Parameterized ODE: If the data is $C^k$, then so is the solution.
I want to prove the following "regularity".
Assume $f \in C^k(\mathbb R^3)$ and $g \in C^k(\mathbb R)$. Consider the parameterized ODE
$$\begin{cases}\displaystyle
\frac \partial {\partial ...
- 622
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0
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39
views
Finite time blow-up for solutions of a quasi-parabolic system
I am reading a book about the Harmonic Map flow and in the proof of the existence of solution the author states something I am not totally familiar with. Let me recall the statement of the theorem.
...
- 3,898
0
votes
1
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123
views
Regularity on the Poisson equation on bounded domains
This is a follow-up question to Regularity of solution for Poisson equation on bounded domains.
We consider the Poisson equation $-\Delta u=f$ in $\Omega$ and $u=\varphi$ on $\partial\Omega$.
...
- 828
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1
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Can we prove elliptic regularity by reducing to a first order system of PDE?
Theorem 10.3.6 of Nicolaescu's notes prove elliptic regularity for a first order PDO. I noted that this PDO is allowed to be interpreted as a system of first order pde's. Now, given any linear, ...
- 7,298
0
votes
1
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66
views
variational form of fourth order elliptic equation
Given
$$
\alpha\Delta^2y+y=\alpha^{1/4}y_d-\alpha^{5/4}\Delta(f+u_d), \Omega
$$
$$
y=0, \alpha^{1/2}\Delta y+\alpha^{3/4}(f+u_d)=0
$$
I am trying to verify the regularity result as
$$
\alpha^{1/2}|y|_{...
- 135
0
votes
1
answer
114
views
Regularity of solution for Poisson equation on bounded domains
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain and consider $-\Delta u=f$ in $\Omega$, $u=\varphi$ on $\partial\Omega$.
For balls with some radius the following result is known: Let $\varphi \...
- 828
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1
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63
views
Can first order linear PDE be elliptic?
In Corollary 3.1.15 of Hormander's Analysis of Linear Partial Differential Operators, he proves that if a distribution $u \in \mathcal D'(\Bbb R)$ satisfies $u' + a u = f \in C(\Bbb R)$, then $u \in C^...
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0
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Cauchy Problem as Continuous Operator between Normed Spaces
Let $\lambda \colon [0, T] \times (0, \infty) \to [0, \infty)$ as well as some reasonable $f \colon (0, \infty) \to [0, \infty)$ and consider the following Cauchy boundary value problem:
$$\partial_t ...
- 1,200
2
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0
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97
views
Regularity of heat equation with nonhomogeneous boundary conditions
Consider the heat equation
\begin{equation*}
\begin{cases}
u_{t} - \Delta u = f &\text{in }(0,T)\times\Omega\\ u(x,t) = g(x,t) &(t,x)\in [0,T]\times\partial\Omega\\ u(x,0) = h(x) &\text{on ...
- 1,295
5
votes
0
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74
views
PDE with a non-classical boundary condition
Assume that one has a classical PDE, say: $u_t(t,x)-u_{xx}(t,x)=0$, $t\in (0,1)$, $x\in (0,2)$, and $u(0,x)=0$. Then we can prove existence (and uniqueness) of solution when boundary conditions: $u(t,...
- 271
2
votes
1
answer
138
views
How to understand the changes of characteristics when we convert a high order PDE to first order PDEs?
Consider a second order PDE, \begin{eqnarray*}
au_{tt}+bu_{tx}+cu_{xx} & = & 0,
\end{eqnarray*} which is equivalent to the following first order PDEs (introducing new variables $p$ and $q$)
\...
- 21
2
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1
answer
133
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Regularity of Elliptic partial differential equation with mixed Dirichlet-Robin boundary condition, to prove $u\in H^{2}(\Omega)$
When I deal with the wave equation with dynamical boundary condition, I am confused by the regularity of the following elliptic partial differential equation with mixed Dirichlet-Robin boundary ...
1
vote
1
answer
27
views
Proving a regularity estimate for $−εu′′ + bu′ = f $
I've been given:
$$−εu′′ + bu′ = f ,x ∈ (0, 1), u(0) = u(1) = 0$$
and have been asked to prove the regularity estimate
$$∥u′′∥_{L^2(0,1)} ≤ C_R∥f∥_{L^2(0,1)}$$
I normally try to provide some working ...
- 305
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0
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58
views
Can you bound the H1 norm of the solution of a Poisson problem with its L2 norm?
Let $u \in H^1(\Omega)$ be such that
$$
\int_{\Omega} \nabla u \cdot \nabla v + \int_{\Omega} u v
= \int_{\partial\Omega} q v
\qquad \forall v \in H^1(\Omega)
$$
for a given boundary datum $q \in H^{-...
0
votes
0
answers
22
views
the existence of solution for Helmholtz equation with Dirichlet and Neumann boundary condition outside a bounded domain.
Consider the Helmholtz equation:
$$\Delta u + k^2u = 0 \quad in \quad \mathbb{R}^n\backslash\overline{\Omega} $$
$$u = f , \frac{\partial u}{\partial n} = g \quad on \quad \partial \Omega$$.
where $f,...
- 1
1
vote
0
answers
29
views
Proving existence and regularity of a weak solution by a standard Galerkin scheme?
I am currently reading a paper on higher-order Cahn-Hilliard equations:
Laurence Cherfils, Alain Miranville, Shuiran Peng, Higher-order anisotropic models in phase separation, Advances in Nonlinear ...
- 163
1
vote
0
answers
37
views
Regularity of FEM a priori
I am studying the book Finite Element Analysis by Szabó and Babuska, and specifically, section 1.5.1 on regularity defines
$$
u_{EX} = x^{\alpha} \phi (x), \ \alpha > 1/2, \ x \in I = (0, \ell). \...
- 39
0
votes
0
answers
63
views
$L^{1}$ bounded solution of elliptic equation implies $L^{\infty}$ boundedness
$Lu=\Delta u+hu$ is an elliptic operator with $h$ smooth on $\overline{\Omega}$. $u_{\alpha}$ is a family of solution of $Lu=0$ on a bounded domain $\Omega$.And we know $|\int_{\Omega}u_{\alpha}|\leq ...
- 1,156
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0
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41
views
Sequence of positive solutions of elliptic PDE
Let $\alpha \geq 0$ be a parameter, and suppose for each $\alpha$, we have two functions $v_{\alpha}, w_{\alpha} \in C^2(\mathbb{R}^2)$ (uniformly bounded) such that $w_{\alpha}>v_{\alpha}$ in $\...
- 223
0
votes
0
answers
37
views
Hölder regularity
I have a question about Hölder regularity of functions. That is, if is possible to split the regularity in any variables to prove regularity in each of them.
For example, let us suppose that $u:(0,1)^...
- 41
2
votes
0
answers
31
views
Linear Elliptic Problems: Are gradient estimates preserved after perturbation?
We start with the linear elliptic PDE
$$
-\operatorname{div}(A\nabla u)=f \quad\text{in}\ \Omega,\\
u=0 \quad\text{on}\ \partial\Omega
$$
We assume that $\Omega\subset\mathbb{R}^3$ is a smooth domain, ...
- 428
1
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1
answer
102
views
Regularity of Weak solution for biharmonic problem
Suppose $\Omega\subset \mathbb R^2$ be a bounded convex polygonal domain. $f\in L^2(\Omega)$ be a force function. Then for the problem
\begin{align}
-\Delta u&=f\quad\text{in }\Omega\\
u&=0\...
- 848
1
vote
1
answer
101
views
Harmonic functions are continuous [duplicate]
Are harmonic functions continuous? I mean harmonic in the weak sense that second partial derivatives exist and $\Delta u=0$ on an open neighborhood. Many sources start with the assumption that ...
- 4,534