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Estimates on derivatives of Coulombic wave functions and their electron densities

Søren Fournais ORCID logo and Thomas Østergaard Sørensen ORCID logo EMAIL logo

Abstract

We prove a priori bounds for all derivatives of non-relativistic Coulombic eigenfunctions ψ, involving negative powers of the distance to the singularities of the many-body potential. We use these to derive bounds for all derivatives of the corresponding one-electron densities ρ, involving negative powers of the distance from the nuclei. The results are both natural and optimal, as seen from the ground state of Hydrogen.

Award Identifier / Grant number: EXC-2111-390814868

Award Identifier / Grant number: 202859

Funding source: Det Frie Forskningsråd

Award Identifier / Grant number: DFF-4181-00221

Funding statement: This article may be reproduced in its entirety for non-commercial purposes. Søren Fournais was partially supported by a Sapere Aude Grant from the Independent Research Fund Denmark, Grant number DFF-4181-00221, and by the European Research Council, ERC grant agreement 202859. Thomas Østergaard Sørensen was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy (EXC-2111-390814868).

A Some new a priori estimates

In this appendix we state and prove a few results related to the a priori estimate proved in [28, Theorem 1.2] (see also the discussion in [39, (19.17)]).

We start by recalling that estimate.

Theorem A.1 ([28, Theorem 1.2]).

Let ψ be as in (1.2). For all R(0,), there exists a constant C>0 such that

(A.1)sup𝐲B(𝐱,R)|ψ(𝐲)|Csup𝐲B(𝐱,2R)|ψ(𝐲)|

for all xR3N.

The proof of (A.1) is based on an “Ansatz” (see also (A.8) below) for the solution of the eigenvalue equation, and then on using elliptic regularity on the resulting equation. The objective of this Appendix is the following strengthening of Theorem A.1:

Proposition A.2.

Let H be the operator in (1.1). For all 0<r<R and EC there exists C=C(r,R,E) (depending also on N,Z) such that if Hψ=Eψ, ψWloc2,2(R3N), then

(A.2)ψL(B3N(𝐱0,r))+ψL(B3N(𝐱0,r))CψL2(B3N(𝐱0,R))

for all x0R3N.

Proof.

Define, for 𝐱=(x1,,xN)3N,

(A.3)F~(𝐱)=j=1N(-Z2|xj|+Z2|xj|2+1)
+1j<kN(14|xj-xk|-14|xj-xk|2+1).

Note that there exists a constant C=C(N,Z)>0 such that (for Σ, see (1.3))

(A.4)|F~(𝐱)|,|𝐱F~(𝐱)|Cfor all 𝐱3NΣ.

Next, let (for V, see (1.1))

G~(𝐱)=-[j=1NZ2Δ𝐱(|xj|2+1)-1j<kN14Δ𝐱(|xj-xk|2+1)]
=V(𝐱)-Δ𝐱F~(𝐱).

Since |Δx(|x|2+1)|3 for all x3, there exists a constant C=C(N,Z)>0 such that

(A.5)|G~(𝐱)|Cfor all 𝐱3N.

Therefore, with

(A.6)K~(𝐱):=G~(𝐱)-|𝐱F~(𝐱)|2-E,

using (A.4) and (A.5), there exists a constant C=C(N,Z,E)>0 such that

(A.7)|K~(𝐱)|Cfor all 𝐱3NΣ.

Define

(A.8)ψ~:=e-F~ψ,

then, (using that Hψ=Eψ), ψ~ satisfies the equation

(A.9)-Δ𝐱ψ~-2𝐱F~𝐱ψ~+K~ψ~=0,

with

(A.10)𝐱F~,K~L(3N).

Note that since ψWloc2,2(3N), we have that ψ~Wloc2,2(3N). This follows from (A.4) and Hardy’s inequality (note that any second order derivative of F~ behaves like |xj|-1 and |xj-xk|-1).

It follows from Theorem C.2 in Appendix C that ψ~Cloc1,θ(3N) for all θ(0,1). In particular, since this, (A.4), (A.7), and (A.9) then implies that

-Δ𝐱ψ~=2𝐱F~𝐱ψ~-K~ψ~Llocp(3N)for all p[1,],

it follows from Theorem C.4 that ψ~Wloc2,p(3N) for all p[2,).

It now follows from Theorem C.3 (used on (A.9), with p=2) that for all R~,r~>0 there exists a constant C=C(r~,R~) (depending also on N,Z,E through (A.4) and (A.7)) such that, for all 𝐱03N,

(A.11)ψ~W2,2(B3N(𝐱0,r~))Cψ~L2(B3N(𝐱0,R~)).

Hence, by Theorem C.1 (i) (Sobolev embedding; with k=2, n=3N, and p=p1=2, q=p1*=6N3N-4>2+83N=p1+83N), and then (A.11), there exists a constant C=C(r~,R~) such that

(A.12)ψ~Lp1*(B3N(𝐱0,r~))Cψ~L2(B3N(𝐱0,R~))<.

Now, using Theorem C.3 again, but this time with p=p2=p1*, and then (A.12), we therefore get that, for all r^(0,r~), there exists a constant C=C(r^,r~,R~)>0 such that

ψ~W2,p2(B3N(𝐱0,r^))Cψ~L2(B3N(𝐱0,R~))

with p2>p1+83N=2+83N. (Of course, the constant C changes every time.)

We repeat this: Sobolev embedding, in the form of Theorem C.1 (i) (always with k=2 and n=3N; next time with W2,p2 and Lp2*, p2*>p2+83N), and then Theorem C.3 (with p=p3=p2*) as long as 2pi<3N. Note that 2=p1<p2<p3< with

pi+1=pi*=3Npi3N-2pi>pi+2pi23N>pi+83N,i=1,2,.

Hence, we reach pM satisfying 2pM<3N<2pM* in maximally 3N-28/3N+1=18(9N2-6N+8) steps (that is, M is smaller equal this number). As above, the radius of the smaller ball decreases each time (above, from r~ to r^). However, splitting the original difference R-r2=R-R+r2 in M+1 equally large parts (we use Theorem C.3M+1 times), we get: For all 0<r<R there exists a constant C=C(r,R)>0 such that

(A.13)ψ~W2,pM*(B3N(𝐱0,r+R2))Cψ~L2(B3N(𝐱0,R)),

with 2pM<3N<2pM*.

Now use Theorem C.1 (ii) (Morrey’s Theorem): With k=2,p=pM*,n=3N (so kp>n), to get, for some θ(0,1),

ψ~Cθ(B3N(𝐱0,r+R2)¯)Cψ~W2,pM*(B3N(𝐱0,r+R2)).

Using (A.13), and that ψ~Lψ~Cθ, this implies that, for all 0<r<R,

(A.14)ψ~L(B3N(𝐱0,r+R2))Cψ~L2(B3N(𝐱0,R))

for some C=C(r,R)>0.

Hence, using (A.9)–(A.10), Theorem C.2 (used on (A.9)), and (A.14) give that, for all θ(0,1),

ψ~C1,θ(B3N(𝐱0,r))Cψ~L(B3N(𝐱0,r+R2))Cψ~L2(B3N(𝐱0,R)).

Hence (since ψ~L+ψ~Lψ~C1,θ), (A.2) follows, but with ψ~ instead of ψ. It remains to recall that

ψ=eF~ψ~

(see (A.8)) with F~ (globally) Lipschitz (see also (A.4)), to arrive at (A.2) for ψ. ∎

As a consequence of Proposition A.2, we get the following, which is of independent interest:

Proposition A.3.

For N2, let H be the operator in equation (1.1). Then, for all 0<r<R and all ER, there exists a constant C=C(r,R,E)>0 such that if Hψ=Eψ, ψW2,2(R3N), and if ρ is the associated one-electron density as in (1.27), then, for all x1R3,

(A.15)3N-3ψL(B3N((x1,𝐱^1),r))2𝑑𝐱^1CB3(x1,R)ρ(y1)𝑑y1,
(A.16)3N-3ψL(B3N((x1,𝐱^1),r))2𝑑𝐱^1CB3(x1,R)ρ(y1)𝑑y1,
(A.17)ρ(x1)=3N-3|ψ(x1,𝐱^1)|2𝑑𝐱^1CB3(x1,R)ρ(y1)𝑑y1,

and

(A.18)|ρ(x1)|=|3N-3x1(|ψ(x1,𝐱^1)|2)d𝐱^1|CB3(x1,R)ρ(y1)𝑑y1,

in the sense that, for all vR3, the directional derivative (exists and) satisfies

(A.19)|vρ(x1)|C|v|B3(x1,R)ρ(y1)𝑑y1.

Furthermore, for all b[0,3) and R>0 there exists a constant C=C(b,R,E)>0 such that

(A.20)3N-3|x2|-b|ψ(x1,𝐱^1)|2𝑑𝐱^1CB3(x1,R)ρ(y1)𝑑y1.

Remark A.4.

Note that, for all x13, R>0,

B3(x1,R)ρ(y1)𝑑y1ρL1(3)=ψL2(3N)2<.

In particular, it follows from (A.18) that ρ is globally Lipschitz, namely, ρC0,1(3). This was already known [28, Theorem 1.11 (i)].

Proof.

We start by proving (A.15) and (A.16) from which the other estimates will follow in a simple manner. Using (A.2) and Fubini’s Theorem,

(A.21)3N-3ψL(B3N((x1,𝐱^1),r))2𝑑𝐱^1
C3N-3ψL2(B3N((x1,𝐱^1),R))2𝑑𝐱^1
=C3N|ψ(𝐲)|2({|𝐲-(x1,𝐱^1)|R}𝑑𝐱^1)𝑑𝐲.

Now, for all 𝐲=(y1,𝐲^1)3N,

{|𝐲-(x1,𝐱^1)|R}𝑑𝐱^1𝟙{|y1-x1|R}{|𝐲^1-𝐱^1)|R}𝑑𝐱^1,

and the last integral equals the volume of B3N-3(0,R) for all 𝐲^13N-3. Inserting this in (A.21) and using the definition of ρ in (1.27) finishes the proof of (A.15). The proof of (A.16) is similar.

To prove (A.17), notice that

3N-3|ψ(x1,𝐱^1)|2𝑑𝐱^13N-3ψL(B3N((x1,𝐱^1),R2))2𝑑𝐱^1

and use (A.15) with r=R2.

To prove (A.18), we differentiate and estimate, to get that

(A.22)|x1(|ψ(x1,𝐱^1)|2)|2ψL(B3N((x1,𝐱^1),R2))ψL(B3N((x1,𝐱^1),R2)).

Here (A.22) should be understood in terms of directional derivatives in the same way as in (A.19). From [28, Proposition 1.5] we know that the directional derivatives of ψ exist.

At this point we can use (A.2) and finish the estimate as above.

To prove (A.20), it suffices, using (A.17), to estimate

{|x2|R4}|x2|-b|ψ(x1,𝐱^1)|2𝑑𝐱^1.

We argue in a similar fashion as above, with 𝐱^1,2=(x3,,xN). Since for all |x2|R4, (x1,x2,𝐱^1,2)B3N((x1,0,𝐱^1,2),R2), we get from Fubini’s Theorem and (A.2) that

(A.23){|x2|R4}|x2|-b|ψ(x1,𝐱^1)|2𝑑𝐱^1
3N-6({|x2|R4}|x2|-b𝑑x2)ψL(B3N((x1,0,𝐱^1,2),R2))2𝑑𝐱^1,2
C(b,R)3N-6ψL2(B3N((x1,0,𝐱^1,2),R))2𝑑𝐱^1,2
=C(b,R)3N|ψ(𝐲)|2({|𝐲-(x1,0,𝐱^1,2)|R}𝑑𝐱^1,2)𝑑𝐲.

Here we also used that b[0,3). Now, for all 𝐲=(y1,y2,𝐲^1,2)3N,

{|𝐲-(x1,0,𝐱^1,2)|R}𝑑𝐱^1,2𝟙{|y1-x1|R}{|𝐲^1,2-𝐱^1,2)|R}𝑑𝐱^1,2,

and the last integral equals the volume of B3N-6(0,R) for all 𝐲^1,23N-6. Inserting this in (A.23) and using the definition of ρ in (1.27) finishes the proof of (A.20). ∎

B A partition of unity

In this appendix we gather various facts about a particular partition of unity (on 3N), needed when studying the electron density ρ; see Section 6.

We denote by Cb(Ω) the set of all smooth functions on Ω which are bounded together with all their derivatives.

Let χ1,χ2Cb(), 0χi1, i=1,2, χ1,χ2 both monotone, with

(B.1)χ1(t)={1,t14,0,t34,andχ2(t)={0,t14,1,t34,

and

(B.2)χ1(t)+χ2(t)=1for all t.

The partition of unity depends on an index IX, where X=J=0N-1XJ, with the sets XJ to be described below. Here

XJ=0={(0,{2,,N},)},

and the corresponding function in the partition of unity is (with 𝐱=(x1,,xN)3N),

(B.3)χ(0,{2,,N},)(𝐱)=j{2,,N}χ1(|xj||x1|).

For each J1, the set XJ consists of all elements of the form (J,PJ,QJ-1,,Q0) with Q0,Q1,,QJ-1,PJ{2,,N} disjoint, and PJ(s=0J-1Qs)={2,,N} (possibly PJ= or Qs=,s1). The corresponding function is (with j=1)

(B.4)χI(𝐱)=χ(J,PJ,QJ-1,,Q0)(𝐱)
=[jPJχ1(4J|xj||x1|)][jQJ-1χ2(4J-1|xj||x1|)χ1(4J-2|xj||x1|)]
××[jQsχ2(4s|xj||x1|)χ1(4s-1|xj||x1|)]
××[jQ1χ2(41|xj||x1|)χ1(40|xj||x1|)][jQ0χ2(40|xj||x1|)].

Lemma B.1.

Let χ1 and χ2 be as above satisfying (B.1) and (B.2). Then (as functions of x=(x1,,xN)R3N),

1=IχI,

where the sum is over a subset of X.

Proof.

To ease notation, let, for 𝐱=(x1,,xN)3N,

(B.5)χi,js(𝐱)=χi(4s|xj||x1|),i=1,2,j=2,,N,s=0,1,2,.

Note that, by (B.1), for all j and s=1,2,,

(B.6)χ1,jsχ1,js-1=χ1,js.

Using (B.2), we have (again, with j=1)

(B.7)1=j=2N[χ1,j0+χ2,j0]=p0q0={2,,N},p0q0=[jp0χ1,j0][jq0χ2,j0].

The term in (B.7) with q0= equals

j{2,,N}χ1,j0=χ(0,{2,,N},).

The term in (B.7) with p0= equals

j{2,,N}χ2,j0=χ(1,,{2,,N}).

For all other terms χp0,q0=[jp0χ1,j0][jq0χ2,j0] in (B.7) we have q0p0, and so 0<#p0<#{2,,N}=N-1. In each of these terms, insert a factor of (recall (B.5) and (B.2))

1=jp0[χ1,j1+χ2,j1],

and multiply out, to get

(B.8)χp0,q0=[jp0χ1,j0]1[jq0χ2,j0]
=q1p1=p0,q1p1=[jp0χ1,j0][jp1χ1,j1][jq1χ2,j1][jq0χ2,j0].

By (B.6), χ1,j0χ1,j1=χ1,j1 for all jp1p0, and so, since p0=q1p1, each of the terms in the sum in (B.8) is of the form

(B.9)χp1,q1,q0=[jp1χ1,j1][jq1χ2,j1χ1,j0][jq0χ2,j0].

As before, the term with p1= (that is, q1=p0) equals

[jq1χ2,j1χ1,j0][jq0χ2,j0]=χ(2,,q1,q0)

and the term with q1= (that is, p1=p0) equals

[jp1χ1,j1][jq0χ2,j0]=χ(1,p1,q0).

For the rest of the terms in (B.9), we have q1p1, and so 0<#p1<#p0<N-1, that is, 0<#p1<N-2. For each of these terms χp1,q1,q0 (with p1q1q0={2,,N}, p1,q1,q0 disjoint), insert a factor of

1=jp1[χ1,j2+χ2,j2],

and proceed as above, using (B.6) with s=2, to write χp1,q1,q0 as a sum (over p2,q2 with p2q2=p1, p2q2=) of terms of the form

(B.10)χp2,q2,q1,q0=[jp2χ1,j2][jq2χ2,j2χ1,j1][jq1χ2,j1χ1,j0][jq0χ2,j0].

Again, the terms with p2= or q2= have (see (B.4)) the correct form (namely, with J=3, P3=p2, Qi=qi,i=0,1,2, I=(3,,Q2,Q1,Q0)X3, and, respectively, with J=2, P2=p2,Qi=qi,i=0,1, I=(2,P2,Q1,Q0)X2). Furthermore, for all other terms χp2,q2,q1,q0 in (B.10), we have q2p2, hence, 0<#p2<#p1<N-2, that is, 0<#p2<N-3. Continuing like this, we get a sum of terms of the form in (B.4), with the size of pj diminishing at each step, until #pk=1 (which occurs for k=N-3). Then the above two possibilities – pk= or qk= – are the only two, and we are done. ∎

The localization functions χI above are constructed in order to have the following lemma, bounding certain terms in the Coulomb-potential by |x1|-1, on the support of χI.

Lemma B.2.

Let χ1 and χ2 be as in (B.1)–(B.2), and define χI as in (B.4). Then there exists a constant C=C(N)>0 such that for all x=(x1,,xN)suppχI,

(B.11)|xj|-1C|x1|-1for all jj=0J-1Qj,
(B.12)|x1-xj|-1C|x1|-1for all j(j=1J-1Qj)PJ,
(B.13)|xj-xk|-1C|x1|-1for all jPJ,kj=0J-1Qj.

Proof.

To prove (B.11), note that, since χI(𝐱)0, for all the stated indices j we have

χ2(4s|xj||x1|)0

for some s{1,,J-1}, JN. Hence, by (B.1),

|xj|1414s|x1|1414N|x1|=cN|x1|,

which proves (B.11).

To prove (B.12), note that, for jPJ, we have

χ1(4J|xj||x1|)0.

Hence, by (B.1), |xj|34|x1|, and so |x1-xj|14|x1| for these j.

On the other hand, for jQ1QJ-1,

χ1(4s-1|xj||x1|)0

for some s{1,,J-1} , JN. Hence, by (B.1), we have |xj|3414s-1|x1|34|x1|, and so (B.12) holds also for these indices j.

Finally, to prove (B.13), note that for the stated indices j, we have |xj|3414J|x1|, and for the stated indices k, we have, for some s{1,,J-1},

|xk|1414s|x1|1414J-1|x1|.

Therefore,

|xj-xk||xk|-|xj|
1444J|x1|-3414J|x1|
=1414J|x1|
1414N|x1|=cN|x1|,

which proves (B.13). ∎

Remark B.3.

This last argument is the reason why we need 4J in χ1 in the PJ-factor, and at most 4J-1 in χ2 in the Qs-factors, in (B.4).

The next lemma uses the previous one, to control derivatives with respect to x1 of (a slightly changed version of) the localization functions χI.

Lemma B.4.

Let χ1 and χ2 be as in (B.1)–(B.2), and let χI be as in Lemma B.1. For x=(x1,,xN)R3N, define 𝐱~=(x~1,,x~N) with

x~j={xj,if j=1 or jPJ,x1+xj,𝑒𝑙𝑠𝑒.

Define finally

(B.14)χ~I(𝐱)=χI(𝒙~).

Then, for all βN03, there exists a constant C=C(I,β) such that

(B.15)|(x1βχ~I)(𝐱)|C|x1|-|β|for all 𝐱=(x1,,xN)3N.

Furthermore, if |β|1, then there exist j{2,,N} and a constant C=C(I,β,j) such that, for all x=(x1,,xN)R3N and all nN0,

(B.16)|(x1βχ~I)(𝐱)|C|xj|-n|x1|n-|β|

or

(B.17)|(x1βχ~I)(𝐱)|C|x1+xj|-n|x1|n-|β|.

Proof.

First note that, by Leibniz’ rule, to prove (B.15) it suffices to prove that for all γ03, there exists a constant such that

(B.18)|(x1γf)(x1)|C|x1|-|γ|

for f any of the functions

(B.19)χ1(4k|xj||x1|),χ1(4k|x1+xj||x1|),χ2(4k|x1+xj||x1|),

where k{0,,N},j1. By the choice of χ1 and χ2, the bound (B.18) is trivial for γ=0 (with C=1). In particular, (B.15) trivially holds if β=0 (again, with C=1).

Secondly, note that in each case, for any γ03{0},

(B.20)(x1γf)(x1)=1m|γ|γ1++γs=γcm,γχi(m)(g(x1))(x1γ1g)(x1)(x1γsg)(x1)

with i=1 or 2, and g(x1) either 4k|xj||x1| or 4k|x1+xj||x1|. On supp(χi(m)g) (m1) we have, in all cases (see (B.1)),

(B.21)14g(x1)34.

Hence, if g(x1)=4k|xj||x1|, then for any γ03{0}, on supp(χi(m)g),

(B.22)|(x1γg)(x1)|cγ,k|xj||x1|-1-|γ|c~γ,k|x1|-|γ|.

On the other hand, if g(x1)=4k|x1+xj||x1|, then for any γ03{0}, again on supp(χi(m)g),

(B.23)|(x1γg)(x1)|=|σγ(γσ)(x1σ|x1+xj|)(x1γ-σ|x1|-1)|
σγcγ,σ|x1+xj|1-|σ||x1|-1-|γ|+|σ|
c~γ,k|x1|-|γ|.

In both (B.22) and (B.23), the second inequality follows from (B.21).

Hence, (B.20), (B.22), (B.23), and the fact that all derivatives of χ1 and χ2 are globally bounded, imply that

|(x1γf)(x1)|1m|γ|γ1++γs=γc~m,γ|x1|-|γ1||x1|-|γs|=C|x1|-|γ|.

This finishes the proof of (B.18) in the case |γ|1, and hence the proof of (B.15).

To prove that (B.16) or (B.17) hold when |β|1, notice that in this case at least one of the functions in the product in (B.14) (that is, in (B.19)) gets differentiated (that is, |γ|1). For this one, do as above, but use additionally (B.21) to get, for all n,

|x1|n|xj|nor|x1|n|x1+xj|n

(as before, on supp(χi(m)g)). Applying this in (B.22) or (B.23) yields (B.16) or (B.17). ∎

C Needed a priori estimates

In this final appendix section we collect needed results from the literature. We start by Sobolev embedding.

Theorem C.1 ([7, Theorem 6, p. 284], [1, 4.12 Theorem, p. 85]).

Let ΩRn be open and bounded, and let kN,p1.

  1. Assume Ω satisfies an interior cone condition. Then, for any k,p with kp<n, we have the continuous embedding

    Wk,p(Ω)Lq(Ω) for all q[p,p*] with p*:=npn-kp.

    Moreover, there exists a constant C=C(k,p,n,Ω) such that

    uLq(Ω)CuWk,p(Ω)for all uWk,p(Ω).
  2. Assume Ω is locally Lipschitz. Then for kp>n, we have the continuous embedding

    Wk,p(Ω)Ck-1-[np],θ(Ω¯)for all θ[0,θ0],

    where

    θ0={[np]+1-(np),if np is not an integer,any positive number less than 1,if np is an integer.

    Moreover, there exists a constant C=C(k,p,n,θ,Ω) such that

    uCk-1-[np],θ(Ω¯)CuWk,p(Ω)for all uWk,p(Ω).

Next, we list some results on elliptic regularity.

The following theorem is adapted from [22, Theorem 8.32] by choosing aij=δij and bi=fi=0 for i,j=1,,n.

Theorem C.2 ([22, Theorem 8.32]).

Let θ(0,1), and let uC1,θ(Ω) be a weak solution of

(-Δ+c(x)+d(x))u=g

in a bounded domain ΩRn, with ci,d,gL(Ω), with

maxi=1,,n{ciL(Ω)},dL(Ω)K.

Then, for any subdomain ΩΩ, we have

uC1,θ(Ω¯)C(uL(Ω)+gL(Ω))

for C=C(n,K,d) where d=dist(Ω,Ω).

The following theorem is adapted from [22, Theorem 9.11] by choosing aij=δij for i,j=1,,n.

Theorem C.3 ([22, Theorem 9.11]).

Let Ω be an open set in Rn and suppose that uWloc2,p(Ω)Lp(Ω), 1<p<, is a strong solution of the equation

(-Δ+b(x)+c(x))u=f

in Ω with bi,cL(Ω), fLp(Ω), with

maxi=1,,n{biL(Ω)},cL(Ω)Λ.

Then for any subdomain ΩΩ,

uW2,p(Ω)C(uLp(Ω)+fLp(Ω)),

where C depends on n,p,Λ,Ω, and Ω.

Theorem C.4 ([23, Lemma 2.4.1.4]).

Let Ω be an open and bounded set in Rn, let 2p<, and let uW2,2(Ω) be a strong solution of the equation

-Δu=f

in Ω with fLp(Ω). Then uWloc2,p(Ω).

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Received: 2018-04-13
Revised: 2020-10-02
Published Online: 2021-04-29
Published in Print: 2021-06-01

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