# Seeking for references on some PDEs

This is not a technical mathematical question. I came across some PDEs with no references nor their names.

$$-\Delta u + \int_\Omega udx = f\qquad \hbox{in \Omega} \label{1}\tag{Eq1}$$

The above equation can be augmented either with Dirichlet boundary condition $$u=g$$ on $$\partial\Omega$$ or with Neumann boundary condition $$\partial_n u=g$$ on $$\partial\Omega$$.

I would like to know the name and the application of this type of problem. Or any good reference to this.

A slightly more general setting would be considering $$-\Delta u + \varphi\Big(\int_\Omega udx\Big) = f\qquad \hbox{in \Omega} \label{2}\tag{Eq2}$$ For an appropriate function $$\varphi$$. I have the feeling this must have a good application that because assuming, for instance, $$\varphi=0$$ and \eqref{2} is augmented with the Neumann boundary condition $$\partial_n u=g$$ on $$\partial\Omega$$ then it becomes the classical Neumann problem whose solutions exist if and only if we have the compatibility condition

$$\int_\Omega fdx+ \int_{\partial\Omega} gdx =0.\label{c}\tag{C}$$

In practice, $$g$$ is the flux term, $$f$$ is the source term. I forgot the meaning of the compatibility condition \eqref{c}.

Question 1: What is the name and application of the problem \eqref{1} or \eqref{2}. Or what are references to this problem?

Another problem is the following Dirichlet problem with $$f\in L^p(\Omega)$$ $$1

$$-\Delta u + u = f\qquad \hbox{in \Omega} \label{3}\tag{Eq3}$$

Brezis' book claims there is a unique $$u\in W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega)$$ solving \eqref{3} and we have $$\|u\|_{W^{2,p}(\Omega}\leq c\|f\|_{L^p(\Omega}$$

Question 2: what are the references to this type of problem?

• if you would attempt to solve equation 1 by the finite-element method, discretizing the Laplacian so that it becomes a tridiagonal matrix $M$, then you would have the equation $\sum_{j}(-M+D)_{ij}u_j=f_i$, where $D_{ij}=1$ is the rank-one matrix with all elements equal to unity; then you could call this problem a rank-one perturbation of the Poisson equation. Mar 6, 2021 at 21:23
• @CarloBeenakker: Hmm, I'd argue that (Eq1) itself is a rank-$1$ perturbation of the Possion equation (without any dicretization) since the operator $u \mapsto \int_\omega dx \cdot 1$ is a rank-$1$ operator on $L^p(\Omega)$. Mar 6, 2021 at 21:53
• Just a side question: please do you know the practical meaning of the compatibility condition $\int_Df =0$? I know sometimes $f$ represents the source term and $\nabla u\nu=g, ~~ (g=0)$ stands for the heat flux at the boundary. But I am barely looking for a reference that gives the meaning (in practice or physical ) of the compatibility condition $\int_Df+\int_{\partial D} g =0$ or the meaning (in practice or physical) of the quantity $\int_Df+\int_{\partial D} g$. Mar 8, 2021 at 10:59

Some time ago I've addressed a similar problem in this Q&A, so I feel I can offer something useful regarding the posed question.

Question 1: What is the name and application of the problem \eqref{1} or \eqref{2}. Or what are references to this problem?

The only reference I know that deals extensively with non-local equations i.e. with integrodifferential equations is the monograph of Chia-Ven Pao [2]. Elliptic integrodifferential equations more general than and including \eqref{1} are analyzed in chapter 3, §3.7, pp. 125-133: note that Pao considers the general boundary condition $$\mathbf{B} u(x)\equiv\alpha_0 (x) \frac{\partial u(x)}{\partial n} + \beta_o(x) u(x)= h(x)\qquad x\in\partial\Omega$$ and he develops the theory in Ḧolder spaces, i.e. $$C^{k+\alpha}\equiv C^{(k,\alpha)}$$, $$k\in\Bbb N$$ and $$\alpha\in (0,1)$$. To my knowledge, the equation has not a particular name: however, similar equations are widely used in applied mathematics as, for example, it describes the stationary distribution of neutrons inside a fission reactor in the description of nonlocal combustion problems described in references [A1] and [A3] (cited in [2], chapter 3, §3.8, example 4.c, pp. 137-138; see also other references cited at the end of §3.9, p.138). In nuclear reactor theory, an equation similar to \eqref{1} is obtained in [A2], equation 2.16, p. 51, by integrating the stationary linear Boltzmann equation respect to the velocity direction $$\mathbf{s}\in\Bbb S^3$$: however, the authors do not use the equation in their subsequent development, perhaps because the structure of the integral operator obtained is very complex.

Question 2: what are the references to this type of problem (i.e. \eqref{3})?

Reference [1] chapter IV, §1.2, pp. 172-174, cited in this Q&A, gives a proof of the result cited by Brezis for $$p=2$$ for the more general operator $$\nabla\cdot\big(k(x) \nabla u(x)\big) +a(x) u(x)= f(x).$$ For $$p\neq 2$$, perhaps it is possible to find the result in the classical monograph of Pierre Grisvard, but I am not sure and I should check.

References

[1] V. P. Mikhailov (1978), Partial differential equations, Translated from the Russian by P.C. Sinha. Revised from the 1976 Russian ed., Moscow: Mir Publishers, pp. 396, MR0601389, Zbl 0388.3500. Link on Archive.org.

[2] Chia-Ven Pao (1992), Nonlinear Parabolic and Elliptic Equations, Plenum Press, xv+777, MR1212084, Zbl 0777.35001. Free Springer Link.