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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 $f\in L^p(\Omega)$ $1<p<\infty$

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

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?

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  • $\begingroup$ 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. $\endgroup$ Mar 6 '21 at 21:23
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    $\begingroup$ @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 $L^p(\Omega)$. $\endgroup$ Mar 6 '21 at 21:53
  • $\begingroup$ 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 $. $\endgroup$
    – Guy Fsone
    Mar 8 '21 at 10:59
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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.

Additional references

[A1] J. Bebernes, R. Ely, "Comparison techniques and the method of lines for a parabolic functional equation", (English) Rocky Mountains Journal of Mathematics 12, 723-733 (1982), MR0683864, Zbl 0536.65096.

[A2] Sergio Gallone, Giancarlo Ghilardotti, Fisica dei reattori nucleari: diffusione e rallentamento dei neutroni [Nuclear reactor physics: diffusion and slowing down of neutrons], Saggi scientifici 8, Milano: Feltrinelli, 1964, pp. 303.

[A3] C. V. Pao, "Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory", (English), Journal of Mathematical Analysis and Applications 166, No. 2, 591-600 (1992), MR1160947, Zbl 0762.35049.

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    $\begingroup$ "however, is widely used in applied mathematics as, for example, it describes the stationary distribution of neutrons inside a fission reactor." please any references where such equation is applied or used? Thank for this answer. $\endgroup$
    – Guy Fsone
    Mar 7 '21 at 8:26
  • $\begingroup$ @GuyFsone, I corrected my statement above: I possibly confused two references, in particular another monograph of Antonio Pignedoli dealing extensively with neutron diffusion in the context of parabolic equations (and elliptic equations in the stationary case), however not of nonlocal type. $\endgroup$ Mar 7 '21 at 11:58

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