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I believe the following statement is true.

Let $\Omega$ be a smoothly, bounded domain in $\mathbb{R}^{n}$.

The statement:

Let $u\in H^{1}(\Omega)$ so that

there exists $f\in L^{2}(\Omega) \;s.t.\int_{\Omega}\nabla u\nabla \varphi=\int_{\Omega} f\varphi, \forall \varphi\in H^{1}(\Omega)$.

Then $u\in H^{2}(\Omega)$.


I have searched in the book by Evans and Brezis but not so certain. Could anyone provide a reference for that? It can be seen in Evans's book that $u\in H^{2}_{loc}(\Omega)$.


Thanks so much.

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    $\begingroup$ Well, by clicking the accept buttom, I thought you are satisfied. I don't have the book mentioned in the answer and can't find one in the library, but I have the feeling that $H^2_{loc}$ is the best you could get. $\endgroup$ – Arctic Char Apr 5 at 19:07
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The regularity theorem which states that if $\color{blue}{f\in L^2(\Omega)}$ and $$ \color{blue}{\int_{\Omega}\nabla u\nabla \varphi\,\mathrm{d}x=\int_{\Omega} f\varphi\,\mathrm{d}x\quad \forall \varphi\in H^{1}(\Omega)}\implies\color{blue}{u\in H^2(\Omega)} \tag{PR}\label{pr} $$ is true if and only if some regularity conditions on the boundary of the domain $\partial \Omega$ and on the boundary value of the solution $u$ are assumed (be it a Dirichlet, Neumann or Robin boundary value problem). Otherwise, as @ArcticChar states, you can only expect $u\in H_\mathrm{loc}^2(\Omega)$.
Precisely

  • Valentin Mikhailov, in his textbook [2] at paragraph §2.3 pp. 216-226, proves the $H^k$, $k\in\Bbb N_+$ regularity of the solution of problem \eqref{pr} assuming $\color{red}{\partial\Omega\in C^2}$, $\color{red}{f\in H^k(\Omega)}$ and the following boundary conditions $$ \text{(Dirichlet) } u|_{\partial\Omega}=g\in H^{k+1/2}(\partial\Omega) \quad\text{or}\quad\text{(Neumann)} \begin{cases} \left.\dfrac{\partial u}{\partial \nu}\right|_{\partial\Omega}=g\in H^{k+1/2}(\partial\Omega)\\ \\ \displaystyle{\int_\Omega f(x)\mathrm{d}x=\int_{\partial\Omega} g(x)\mathrm{d}x} \end{cases} $$ and proves the result directly, giving also some indication on how to dealwith the Robin boundary condition.
  • Pierre Grisvard, in his famous monograph [1] at paragraph §2.1 pp. 83-84, states that the solution $u$ of problem \eqref{pr} is $W^{2,p}(\Omega)$, $1<p<+\infty$, provided that $\color{red}{\partial\Omega\in C^{1,1}}$, $\color{red}{f\in L^p(\Omega)}$ and that the following boundary conditions hold $$ \text{(Dirichlet) } u|_{\partial\Omega}\in W^{2-\frac{1}{p},p}(\partial\Omega) \quad\text{or}\quad\text{(Robin) }\,\left.\gamma\left( \frac{\partial u}{\partial \nu}+\sigma u\right)\right|_{\partial\Omega}\in W^{2-\frac{1}{p},p}(\partial\Omega) $$ but does not proves the result directly (he proves it for a more general divergence form operator) and also considers the Neumann boundary condition explicitly only for the Helmoltz operator. Due to this, below we will follow Mikhailov's development.

The first step of Mikhailov is to prove the theorem with homogeneous boundary conditions:

Theorem 4 ([2], p. 217). If $f\in H^k(\Omega)$ and $\partial\Omega\in C^{k+2}$ for certain $k\ge 0$, then the generalized solutions $u(x)$ of the first and second boundary-value problems (respectively called Dirichlet and Neumann problems) with homogeneous boundary condition for the Poisson equation belong to $H^{k+2}(\Omega)$ and satisfy (in the case of the second boundary problem it is assumed that $\int_\Omega u \mathrm{d}x=0$) the inequality $$ \Vert u\Vert_{ H^{k+2}(\Omega)}\le C\Vert f\Vert_{ H^{k}(\Omega)} $$ where the constant $C>0$ does not depend on f.

Then he uses the regularity theorem 4 above to the case of non homogeneous boundary conditions ([2], p. 226) by reducing to homogeneous ones. He explicitly does this only for the Dirichlet problem as we do, but the same method nevertheless works for the Neumann problem, as shown in this answer. Consider a solution $u(x)$ of the problem Dirichlet problem above and a function $\Phi\in H^{k+2}(\Omega)$ whose trace on $\partial\Omega$ is $g\in H^{k+1/2}(\partial\Omega)$, i.e. $$ \Phi=g\;\text{ on }\;\partial\Omega\label{1}\tag{1} $$ Define $w=u-\Phi$: then, for every $\varphi\in H^{1}_0(\Omega)$, $$\label{GeneralizedDirichlet}\tag{GN} \begin{split} \int_{\Omega} \nabla w \cdot \nabla \varphi \, \mathrm{d}x&=\int_{\Omega} \nabla u \cdot \nabla \varphi \, \mathrm{d}x - \int_{\Omega} \nabla \Phi \cdot \nabla \varphi \, \mathrm{d}x \\ &= \int_\Omega f \varphi \, \mathrm{d}x + - \int_{\Omega} \nabla \Phi \cdot \nabla \varphi \, \mathrm{d}x \\ &=\int_\Omega f \varphi \, \mathrm{d}x - \int_{\Omega} \nabla\cdot(\varphi\nabla\Phi) \, \mathrm{d}x + \int_{\Omega} \varphi\Delta\Phi \, \mathrm{d}x \\ &=\int_\Omega \big[\,f + \Delta\Phi\big]\varphi\, \mathrm{d}x , \end{split}\label{2}\tag{2} $$ thus we get $$ \int_{\Omega} \nabla w \cdot \nabla \varphi \, \mathrm{d}x=\int_\Omega f_1 \varphi \, \mathrm{d}x $$ where $f_1=f+\Delta\Phi\in H^k(\Omega)$. This means that $w$ solves the homogeneous Dirichlet problem with homogeneous boundary conditions (i.e. $g\equiv 0$), and by applying theorem 4 we have $$ w=u-\Phi\in H^{k+2}(\Omega) \iff u=w+\Phi\in H^{k+2}(\Omega) $$

Notes

  • In Mikhailov's notation, $H^0=L^2$: also he does not use the standard notation for traces, for example expressing that $\varphi\in H^{1/2}(\partial G)$ by saying that $\varphi$ is a function in $L^2(\partial G)$ which is the trace of a function $\Phi\in H^1(G)$.
  • Why it is necessary to consider first the homogeneous boundary value problem and then reducing the non homogeneous one to it? Possibly because of the Prym-Hadamard phenomenon, i.e. because there exists functions $g\in C^0(\partial\Omega)$ which are not extendible as $\Phi\in H^1(\Omega)$
  • As stated above, assuming $f\in L^2(\Omega)\cap H^{k}_\mathrm{loc}(\Omega)$ (i.e. $f\in L^2(\Omega)$ tout court if $k=0$ is assumed) you can drop any requirement on the boundary of $\Omega$ and on the boundary values of $u$: however the regularity result holds only locally i.e. $u\in H_\mathrm{loc}^{k+2}(\Omega)$ ([2], p. 216). There is also a counterexample to global regularity higher than $H_\mathrm{loc}^{2}(\Omega)$ involving $f\in C^0(\overline\Omega)$ ([2], pp. 246-247): precisely, for $n=2$, $\overline{\Omega}=B(0,R)=\{x\in\Bbb R^2||x|\le R<1\}$ let $$ f=\frac{x_1^2-x_2^2}{2|x|^2}\left(\frac{4}{(-\ln|x|)^{\frac{1}{2}}}+\frac{1}{2(-\ln|x|)^{\frac{3}{2}}}\right)\in C^0(\Omega) $$ Then the solution is such that $u\in H_\mathrm{loc}^{2}(\Omega)$ but $u\notin C^2(\Omega)$ (perhaps there is a typo in the expression of $f$ reported in the book: however, I've not checked all the calculations).
  • Regarding the third boundary problem (the so called Robin problem), Mikhailov ([1], footnote *, p. 217) remarks that analysis of the regularity of solutions can be dealt as done in theorem 4 above for the solutions to the first and second boundary value problems, provided certain conditions are assumed.

[1] P. Grisvard (1985), Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, 24. Pitman Advanced Publishing Program. Boston-London-Melbourne: Pitman Publishing Inc., pp. XIV+410, MR0775683, Zbl 0695.35060.

[2] V. P. Mikhailov (1978), Partial differential equations, Translated from the Russian by P.C. Sinha. Revised from the 1976 Russian ed., Moscow: Mir Publishers, p. 396 MR0601389, Zbl 0388.3500.

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  • $\begingroup$ My question: Let $\Omega\subset \mathbb{R}^{n}$ be a smoothly, bounded domain. Let $u\in H^{1}(\Omega)$ and $f\in L^{2}(\Omega)$ so that $\int_{\Omega}\nabla u\nabla\varphi=\int_{\Omega}f\varphi,\forall \varphi \in H^{1}(\Omega)$. Can we claim that $u\in H^{2}(\Omega)$? It seems that your answer is: NO.\\ Note that if $u$ and $f$ are as in our assumption then we must have $\int_{\Omega}f=0$ (take $\varphi=1$). We may deduce more things from the assumption. However, the key question here is that whether $u\in H^{2}(\Omega)$. I think the answer is YES. $\endgroup$ – Hahn Apr 11 at 2:17
  • $\begingroup$ Your answer above doesn't show any counterexample. The reason I think the answer is YES is that we can apply the whole machinery as in the proof of the Dirichlet case to our problem. Let me recall the Dirichlet case: if $\Omega\in C^{\infty}$, $u\in H^{1}_{0}(\Omega)$, $f\in L^{2}(\Omega)$ and $\int_{\Omega}\nabla u\nabla \varphi=\int_{\Omega}f\varphi,\forall \varphi\in H^{1}_{0}(\Omega)$, then we can claim $u\in H^{2}(\Omega)$. $\endgroup$ – Hahn Apr 11 at 2:26
  • $\begingroup$ @Hahn my answer is not no: my answer is that if you assume a sufficient smoothness of the boundary $\partial\Omega$ of the given domain $\Omega$ the answer is yes and the proof is matter of standard textbooks, for example theorem 4 in Mikhailov stated above. Otherwise you can expect only $u\in H^2_\mathrm{loc}(\Omega)$. $\endgroup$ – Daniele Tampieri Apr 11 at 5:40
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    $\begingroup$ @Danieie When I wrote "smoothly bounded", I meant $\partial\Omega\in C^{\infty}$. You wrote "or more precisely $u\in H^{2}_{0}(\Omega)$", I think we only have $u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$. $\endgroup$ – Hahn Apr 11 at 6:30
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    $\begingroup$ In (PR) do you intend to test with only all $\varphi \in H^1_0(\Omega)$? Testing with all $\varphi \in H^1(\Omega)$ you do impose a zero Neumann condition, so you would get $u \in H^2(\Omega)$ provided $\partial\Omega$ is sufficiently regular (which is what Hahn is asking I think). $\endgroup$ – ktoi Apr 12 at 19:49
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Showing boundary regularity for the Neumann problem is technical, but fundamentally it's not any different from the Dirichlet case. I admittedly haven't checked the details fully, but the main steps are the following (this roughly mirrors Evans' proof of boundary Dirichlet regularity, namely theorem 4 in section 6.3):

  1. By localising about each point on the boundary, assume our domain is the half-ball $\widetilde\Omega = B^n_1 \cap \mathbb R^n_+$ and that $u$ satisfies, $$ B[u,v] = \int_{\widetilde\Omega} fv, $$ for all $u \in H^1(\widetilde\Omega)$ such that $u = 0$ on $(\partial B_1^n) \cap \mathbb R^n_+,$ where $B[\cdot,\cdot]$ is the bilinear form associated to some elliptic operator. Note that $u$ satisfies zero Dirichlet boundary on the curved part of the boundary, and zero Neumann boundary on the flat part. Also $B[\cdot,\cdot]$ is a general elliptic operator now, due to the extra terms arising from flattening the domain.

  2. By a difference quotient argument as in the Dirichlet boundary case, we deduce that $D_kDu \in L^2(\widetilde\Omega')$ for all $1 \leq k \leq n-1,$ where $\widetilde\Omega' = B_{1/2}^n \cap \mathbb R^n_+.$ The idea is to test with tangential difference quotients, noting they are admissible in our test space. It's a good exercise to check things fully here, as you'll see that to justify the same test function works you need to use the boundary condition.

  3. The final derivative $D_{nn}u$ can be estimated in $L^2(\widetilde\Omega')$ by using the equation, as in the Dirichlet case.

  4. Now patch together using a partition of unity argument, to deduce that $u$ is in $H^2(\Omega).$

A couple of remarks:

  • The important point is that when you localise, you no longer have a Neumann boundary condition. Instead you have a mixed condition, and although it doesn't cause any complications you should be aware of this.

  • There are certain conditions one needs for the equation $-\Delta u = f$ with zero Neumann boundary to be solvable, but the regularity theory does not care about this. This is essentially because the necessary conditions are global, while the regularity theory is local.

  • Note that if you keep track of the estimates, you will be able to show that, $$ \lVert u\rVert_{H^2(\Omega)} \leq C\left( \lVert f\rVert_{L^2(\Omega)} + \lVert u\rVert_{L^2(\Omega)}\right).$$ By using the Neumann boundary data you can obtain an estimate of the form $\lVert u - (u)_{\Omega} \rVert_{L^2(\Omega)} \leq \lVert f\rVert_{L^2(\Omega)}$ where $(u)_{\Omega}$ is the average of $u$ on $\Omega$ (sketch: test the equation against $v = u - (u)_{\Omega}$ and apply the Poincaré inequality in 5.8.1 of Evans).

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  • $\begingroup$ How do you know: $u$ satisfies zero Dirichlet boundary on the curved part of the boundary, and zero Neumann boundary on the flat part ? $\endgroup$ – Hahn Apr 7 at 15:02
  • $\begingroup$ The zero Dirichlet on the curved boundary isn't too important in hindsight, but the construction I had in mind was to cut off $u$ away from the boundary (which you do if you take a partition of unity). Here I'm ambusing notation by replacing $u$ by what actually is $(u\eta) \circ \Phi$ for a suitable cutoff $\eta$ and coordinate change $\Phi.$ Checking the Neumann condition should be a similar change of variables argument as in Evans, but again I haven't checked the details. $\endgroup$ – ktoi Apr 7 at 15:08
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Please check a general elliptic regularity result ($L^p$ version) on:

Dauge, Monique. Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions, 1988

Theorem 20.10 (together with the explanation of notation above the theorem), essentially we have the following regularity result: $$ \|u\|_{H^2} \leq C \|\Delta u\|_{L^2}. $$ i.e., in your case, $\Delta u = f$ which is implied the fundamental theorem of calculus of variation. For a more specific version in $L^2$ sense, please refer to Grisvard's book Elliptic Problems in Nonsmooth Domains, $\S 2.3.3$, where the whole chapter 2 deals with weak solutions.

BTW: please be careful with the test function you choose, if your test function is in $H^1(\Omega)$, not in $H^1_0(\Omega)$, this is a Neumann problem and your $u$ and $f$ should satisfy certain compatibility condition.

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  • $\begingroup$ Thank for your answer. I agree that $u$ and $f$ should satisfy certain conditions, such as $\partial_{v} u=0$, $\int f =0$. However, in my question I just ignore these and assume that we are given such functions $u$ and $f$, for simplicity. The book by Dauge, Monique is written in a quite old-fashioned way. It's a bit hard to follow the statement. I am quite surprised that there are not many references for the statement in my question. $\endgroup$ – Hahn Apr 4 at 0:21
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    $\begingroup$ What is your condition on $u$? That inequality seems impossible to satisfies for all $u$, when one can change $u$ to $u+C$. $\endgroup$ – Arctic Char Apr 4 at 0:27
  • $\begingroup$ @ Char/ I don't understand your question. I let $u$ be a function satisfying $u\in H^{1}(\Omega)$ and there exists $f$ .... That is $u$ is given. As I have explained, not any $u\in H^{1}$ satisfies these, however, the question is: given $u$ satisfying these conditions, prove that $u\in H^{2}$. $\endgroup$ – Hahn Apr 4 at 0:31
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    $\begingroup$ The question is directed to Shuhao , @Hahn. I am questioning the condition on $u$ so that $L^2$-estimates holds. Note that if $u$ satisfies the condition in your question, then so is $u+C$ for any $C$. So I assume that inequality cannot be true for all your possible $u$. $\endgroup$ – Arctic Char Apr 4 at 0:33
  • $\begingroup$ @ArcticChar $u$ either has zero Dirichlet BC or has zero average in $\Omega$. $\endgroup$ – Shuhao Cao Apr 4 at 5:45

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