Regularity of Weak Solutions of Elliptic and Parabolic Equations with Some Critical or Supercritical Potentials

0.25 MB·

by  Zijin Li

Regularity of Weak Solutions of Elliptic and Parabolic Equations
with Some Critical or Supercritical Potentials
ZijinLi1, Qi S. Zhang2
6
1
0
2

n
a Abstract
J
WeproveHo¨ldercontinuityofweak solutionsoftheuniformlyellipticandparabolicequations
1
1
A
P] ∂i(aij(x)∂ju(x))− x 2+βu(x) = 0 (A > 0, β ≥ 0), (0.1)
| |
A
. A
h ∂ (a (x,t)∂ u(x,t)) u(x,t) ∂ u(x,t) = 0 (A > 0, β 0), (0.2)
t i ij j − x 2+β − t ≥
a | |
m withcriticalorsupercritical0-ordertermcoefficientswhicharebeyondDeGiorgi-Nash-Moser’s
[ Theory. Wealso prove,in somespecial cases, weak solutionsare evendifferentiable.

1 Previously P. Baras and J. A. Goldstein [3] treated the case when A < 0, (a ) = I and
ij
v
β = 0 for which they show that there does not exist any regular positive solution or singular
5
2 positive solutions, depending on the size of A . When A > 0, β = 0 and (a ) = I, P. D.
ij
| |
3 Milmanand Y. A. Semenov[7][8]obtainboundsfortheheat kernel.
2
0 Keywords: weak solutions,elliptic,parabolic,Ho¨ldercontinuity,critical, supercritical
.
1 potential
0
6
1 2010Mathematical Subject Classification: 35D30,35J15,35K10
:
v
i
X 1. Introduction
r
a
Inthispaper,weconsiderregularityofweaksolutionsofdivergenceformellipticequations
A
∂ (a ∂ u) u = 0 (1.1)
i ij j − x 2+β
| |
and parabolicequation
A
∂ (a ∂ u) u ∂ u = 0 (1.2)
i ij j − x 2+β − t
| |
1Department of Mathematics and IMS, Nanjing University, Nanjing 210093, China.
([email protected])
2DepartmentofMathematics,UniversityofCaliforniaRiverside,CA92521,USA.([email protected])
PreprintsubmittedtoElsevier
2
in the unit ball B := B(0,1)(or B R ) in Rd, with d 3, A > 0, β 0. Here a
+ ij
× ≥ ≥ ∈
L (B) (or L (R ,L (B))), and the second order coefficient matrix a satisfies
∞ ∞ + ∞ ij
1 i,j d
theuniformlyellipticcondition: (cid:16) (cid:17) ≤ ≤
λI a ΛI,forsome 0 < λ Λ < . (1.3)
ij
≤ 1 i,j d ≤ ≤ ∞
(cid:16) (cid:17) ≤ ≤
Here and below, we use the Einstein summation convention. We say u H1(B) is a weak
∈
solutionoftheellipticequationof(1.1), if ψ C (B),thereholds
∀ ∈ 0∞
A
a (x)∂ ψ(x)∂ u(x)dx+ u(x)ψ(x)dx = 0, (1.4)
ij i j x 2+β
ZB ZB | |
where ∂ indicates ∂ here and below. Similarly, for the parabolic equation (1.2), we say
i xi
u L2([0,T],H1(B)) isaweak solution,if ψ C (B [ T,T]) , thereholds
∈ 0 ∀ ∈ 0∞ × −
T T
u(x,t)∂ ψ(x,t)dx+ a (x,t)∂ ψ(x,t)∂ u(x,t)dxdt
t ij i j
−
Z0 ZB Z0 ZB (1.5)
T A
+ u(x,t)ψ(x,t)dxdt = ψ(x,0)u(x,0)dx.
x 2+β
Z0 ZB | | ZB
In the middle of the last century, De Giorgi[5], Nash[11] and Moser[9][10] developed new
methods on the studying of elliptic and parabolic equation, which opened a new area on the
studyofregularityofweak solutionsofellipticand parabolicequationsindivergenceform:
∂ (aij∂ u)+bi∂ u+cu = f +∂ fi, (1.6)
i j i i
∂ (aij∂ u)+bi∂ u+cu ∂ u = f +∂ fi. (1.7)
i j i t i
−
They proved that, under certain integrable conditions of the coefficients bi, c and non-
homogeneoustermf andfi,weak solutionsofequation(1.6),(1.7)haveCα Ho¨ldercontinuity.
A key condition for their theory for elliptic equation is that the coefficient of the 0-order term
c must belong to the Lebesgue space Lp, with p > d. Obviously, the 0-order terms in our
2
equations (1.1) and (1.2) do not satisfy this assumption. Actually the case when β = 0 is the
critical borderline case where the theory of De Giorgi, Nash and Moser fails. See for example
of Baras and Goldstein [3] in the case A < 0. When A > 0, even though it is easily seen that
weak solutionsarelocally bounded,it isnotclear theseweak solutionshaveanyregularity.
However, such equations are closely related to several physical equations. For instance the
3
3-dimensionalaxially-symmetricincompressibleNavier-Stokesequationsinfluiddynamicsare
∂ vr +b vr (vθ)2 +∂ p = ∆ 1 vr
t ·∇ − r r − r2
 ∂ vθ +b vθ + vrvθ = ∆ (cid:16)1 vθ (cid:17)
 t ·∇ r − r2 (1.8)

 ∂ vz +b vz +∂ p = ∆(cid:16)vz (cid:17)
 t z
b = vrr ·+∇vze , b = ∂ vr + vr +∂ vz = 0
r z ∇· r r z



where 
v(x,t) = vr(r,z,t)e +vθ(r,z,t)e +vz(r,z,t)e . (1.9)
r θ z
Observe that the linear parts of the first and second equation of (1.8) are related to equation
(1.2)withβ = 0.
We point out here that the case when A < 0 was studied in [3] Baras and Goldstein, who
provedthattheCauchyproblemofheat equation
∆u(x,t) A u(x,t) ∂ u(x,t) = 0, (x,t) Rd R
− x2 − t ∈ × + (1.10)
| |
u(x,0) = u
(cid:26) 0
2 2
have no weak nonnegative solution if A > d 2 . They also prove if 0 < A d 2 ,
− −
− 2 − ≤ 2
(1.10)hasunboundedpositiveweak solutions(cid:16). (cid:17) (cid:16) (cid:17)
Thecase whenA > 0,β = 0 and theleadingoperatorbeing theLaplacian was first studied
in Milman and Semenov [7], where the authors obtained a sharp upper bound for the funda-
mentalsolutionof(1.2). Theirmethodistouseexplicitspecialsolutionsoftheellipticequation
as weights and convert the studying of the problem to that of a weighted equation via Doob’s
transform. Based on this bound, in this special case, one can prove Ho¨lder continuity of so-
lutions quite easily. Here we observe further that when A is sufficiently large, weak solutions
are even differentiable. In the variable coefficient case that we are working on, it is hard or
impossibleto find an explicit solution of the elliptic equation. So a different method is needed
toproveHo¨ldercontinuityofweak solutions.
The following are the main results of the paper. The first one pertains the elliptic equation
in the case when the leading operator is the Laplacian but the result is stronger, including dif-
ferentiability in some situations. This is a littleunexpected since it is well known that singular
potential terms usually mess up the derivative bound for solutions. We also obtain a similar
result for the corresponding parabolic cases with β = 0. The second result deals with both
elliptic and parabolic equations whose leading coefficients are just bounded. We prove Ho¨lder
continuityofweak solutions.
Theorem 1.1. A weak solution u = u(x) H1(B) of ∆u A u = 0 has the following
∈ 0 − x2+β
| |
regularityproperties. Letα = α(A) = d+2+√d2 4d+4+4A ]n,n+1]foranonnegativeinteger
− −
2 ∈
n.
(I) If β = 0,then u Cn,(α(A) n)− B ;
− 1/4
∈
(cid:0) (cid:1)
4
(II) If β > 0,then u C B .
∞ 1/4
∈
In addition,
(cid:0) (cid:1)
(III) A weak solution u = u(x,t) L2([0,T],H1(B)) of ∆u A u ∂ u = 0 satisfies
∈ 0 − x2 − t
∂m1 m2u C(α(A) n)−;(α(A) n)−/2 B [t ,T] , for2m +m =|n|.
t ∇x ∈ − − 1/4 × 0 1 2
Here,theHo¨ldernormsabovedependond,AandtheL2 normofu. T,t aregivenpositive
0
(cid:0) (cid:1)
constants,and C defines anynumbersmallerthanbutclosetotheconstantC.
−
(cid:3)
Theorem 1.2. If β = 0,theweaksolutionuof theellipticequation(1.1)isHo¨ldercontinuous,
i.e.
u C(λ,Λ,d,A) u (1.11)
k kCα(B1/4) ≤ k kL2(B)
withα = α(λ,Λ,d,A) > 0. Moreoveranyweaksolutionu = u(x,t)oftheparabolicequation
(1.2)isHo¨ldercontinuouswhen tisawayfrom0, i.e.
kukCα;α2(B1/4×[t0,T]) ≤ C(λ,Λ,d,A,)t0−(d/2+1+α)kukL2 (1.12)
with α = α(λ,Λ,d,A) > 0,t > 0.
0
(cid:3)
This theorem provides an interior estimate which deteriorates near initial time. However,
thisisnecessary sinceno Ho¨lderregularityassumptionontheinitialdatumis made.
WealsomentionthattheHarnackinequalitycouldnotholdforsolutionsoftheseequations,
becauseonecanfindaclassofnon-negativesolutionswhichdonotsatisfyit. Seesection2e.g..
Moreover,thesespecial solutionsareinstrumentalin studyingtheregularityofthesolutionsof
A
∆u u = 0, (β 0), (1.13)
− x 2+β ≥
| |
theellipticcasewheretheleadingoperatoristheLaplacian. Ithelpstoproveanα orderdecay
−
estimateoftheweak solutionu = u(x) when β = 0at 0 Rd, namely:
∈
u(x) C x α (1.14)
| | ≤ | |
for α (0,1) and C R is a constant. When β > 0, the decay of u(x) at 0 Rd turns
+
∈ ∈ ∈
exponential.
For the variable 2nd order coefficients case (1.1), (1.2), the situation is more complicated.
Roughly speaking, we could not find a good enough special solution as the Laplacian case
(1.13). However, if β = 0, we find a weighted mean value inequality, which is motivated by
[13] and [12]. The weight, decaying at certain rate near the origin, plays the same role as the
special solutionintheLaplaciancase (1.13), givingsimilarα decay estimate.
−
The rest of the paper is organized as follows. In section 2, we give the proof of Theorem
1.1. In section 3, we state and prove the aforementioned weighted mean value inequality for
5
general parabolic equations (1.2). In section 4, we give the proof of the variable coefficient
case for weak solutions of (1.1) with critical 0-order term coefficient. Finally in section 5, we
extend our conclusion in section 4 to the parabolic case. Some elementary but useful works,
giving the proof of the existence of weak solutions, local boundedness of weak solution, max-
imum principle, and an introduction of the modified Bessel’s equation, could be found in the
Appendix.
2. LaplacianCase
Inthissection,wewillproveTheorem1.1. Firstweneedasimplelemmaoncertainspecial
solutions of (1.13), which will serve as a benchmark for comparison with other solutions. As
mentionedin theintroduction,thecase when β = 0, Ho¨ldercontinuityof solutionscan alsobe
provenbytheboundin[7]. Here wegiveadirect proofbasedon themaximalprinciple.
Lemma 2.1. (i)If β = 0,then
d+2+√d2 4d+4+4A
u(x) = x α, α = α(A) = − − (2.1)
| | 2
isa weak solutionof (1.13).
(ii)Ifβ > 0, then
2
u(x) = x −d2+1 (d 2)/β √A x −β2 (2.2)
| | K − β | |
is a weak solution of (1.13), where is the m(cid:0)odified Bes(cid:1)sel’s function of second kind
(d 2)/β
K −
mentionedabove.
Proof. Since we are looking for radially symmetric solution of (1.13) here, we can just solve
thecorrespondingODE.Define r = x , wefind thesolutionu = u(r). Thus,(1.13)turnsto
| |
d 1 A
u + − u u = 0 (2.3)
′′ ′
r − r2+β
Ifβ = 0, thisisan EulertypeODE. Set u = rα,and takethisintotheequation(2.3), wehave
α2 +(d 2)α A = 0 (2.4)
− −
Solvethisequationwithapositivenumber,wehave
d+2+√d2 4d+4+4A
α = α(A) = − − . (2.5)
2
If β > 0, suppose (r)satisfies themodifiedBessel’s equation
λ
B
r2 (r)+r (r) (r2 +λ2) (r) = 0. (2.6)
Bλ′′ Bλ′ − Bλ
6
Change r into ν rµ, where ν = 0 and µ = 0 are real numbers to be determined later, we have
· 6 6
g(r) := (ν rµ)satisfies thefollowingequationby direct calculation:
λ
B ·
1 µ2ν2 µ2λ2
g + g g g = 0. (2.7)
′′ ′
r − r2 2µ − r2
−
Observe(2.3), wechooseµ = β and ν = 2√A, thus(2.7)becomes
−2 β
1 A βλ
g + g g ( )2g = 0. (2.8)
′′ ′
r − r2+β − 2r
Now,toeliminatethelasttermandmodifythecoefficientofthesecondtermontheleftof(2.8),
weset h(r) := rθ g(r),where θ isa real numbertobedeterminedlater. By directcalculation,
·
wehavehsatisfies
1 2θ A βλ
h + − h h+ θ2 2 r 2h = 0. (2.9)
′′ ′ −
r − r2+β − 2
(cid:2) (cid:0) (cid:1) (cid:3)
Compare(2.9)to(2.3), wehave
θ2 βλ 2 = 0
− 2
(2.10)
 (cid:0) (cid:1)
2θ +1 = d 1

− −
Thus wehaveθ = d +1, and λ =d 2, which we haveh(r) = r d+1 (2√Ar β) solves
−2 −β −2 B2−βd β −2
(2.3). Sinceweare lookingforlocal boundedsolution,wechoose
(r) = (r), (2.11)
d−2 d−2
B β K β
themodifiedBessel’sfunctionofsecond kind,whichis exponentiallygrowingat 0 Rd.
∈
(cid:3)
As for(ii)inLemma2.1 , wehave
Lemma 2.2. The function x d+1 2√A x β is smooth in B, and decays exponen-
| |−2 K(d−2)/β β | |−2
tiallyto0 atx = 0.
(cid:0) (cid:1)
This is a direct corollary of the property of modified Bessel’s function, see [1] for more
details.
(cid:3)
Nowwestart theproofofTheorem 1.1, Case(I).
7
We denoteby J = J (x) thespecialsolutionsof(1.13)mentionedabove,namely
β β
x d+1 2√A x β , β > 0
| |−2 K(d−2)/β β | |−2
J (x) = (2.12)
β  (cid:0) (cid:1)
 −d+2+√d2−4d+4+4A
 x , β = 0.
2
| |

Then,onB ,thereexistsaconstantC = C d, u ,suchthatthefunctionsv = u(x)+
1/2 L2(B) 1
k k
CJ (x) and v = u(x) CJ (x) satisfy
β 2 β
− (cid:0) (cid:1)
∆v (x) 1 v (x) = 0, in B , i = 1,2,
i − x2+β i 1/2
| | (2.13)

v (x) 0, v (x) 0, on ∂B .
 1 2 1/2
≥ ≤
By themaximumprincipleinLemma6.2intheAppendixe.g., wehave
C J (x) u(x) C J (x). (2.14)
β β
− · ≤ ≤ ·
According totheGreen’s Representationformula(c.f.[4]), wehave, forx B
1/2
∈
1
u(x) = Γ(x y) u(y)dy+H(x) (2.15)
− y 2+β
ZB1/2 | |
owfhde-rdeiΓm(exn−siyo)na=l ud|nx(2−i−tydb|2)a−ωlddl)i,satnhdefHund=amHe(nxt)alissohluartimononoifcthineBLapl.acSeinecqeuathtieonse(cωodnidsttheermvoHlu(mxe)
1/2
of(2.15)is regularenough inB , weonlyneed toconsidertheregularityof
1/4
1
w(x) = Γ(x y) u(y)dy. (2.16)
− y 2+β
ZB1/2 | |
Wedividetherest oftheproofintoseveralcases, firstly:
8
2.1. Case(I)β = 0, α(A) 1
≤
By (2.14), x ,x B , definez = 1(x +x ), δ = x x .
∀ 1 2 ∈ 1/4 2 1 2 | 1 − 2|
1
w(x ) w(x ) C Γ(x y) Γ(x y) dy
| 1 − 2 | ≤ | 1 − − 2 − | y 2 α(A)
ZB1/2 | | −
1
C Γ(x y) dy
≤ | 1 − | y 2 α(A)
ZB1/2TB(z,δ) | | −
1 (2.17)
+C Γ(x y) dy
2
| − | y 2 α(A)
ZB1/2TB(z,δ) | | −
1
+C Γ(x y) Γ(x y) dy
1 2
| − − − | y 2 α(A)
ZB1/2−B(z,δ) | | −
= C(I +I +I )
1 2 3
As toI , for d < p < d ,by Ho¨lderinequality
1 2 1 2 α(A)
−
I1 C x1 y (2p−1d−)1p1dy 1−1/p1 y (α(A)−2)p1dy 1/p1
≤ | − | · | |
(cid:16)ZB(x1,32δ) (cid:17) (cid:16)ZB1/2 (cid:17)
Cδ2pp11−d 1 p11 2p1 −d −1+1/p1 (2.18)
≤ · (α(A) 2)p +d · p 1
1 1
h − i h − i
d
C δα(A)−, by choosing p .
1
≤ · → 2 α(A)
−
−
(cid:0) (cid:1)
Here and below, we use C to denote an arbitrary number close but smallerthan C. Similarly,
−
we have I satisfies the same estimate. As for I , by mean value inequality, there exists an xˆ
2 3
liesbetween x and x ,and for1 < p < d :
1 2 2 2 α(A)
−
1
I Cδ Γ(xˆ y) dy
3 ≤ |∇ − | y 2 α(A)
ZB1/2−B(z,δ) | | −
Cδ y (1p−2d−)1p2dy 1−1/p2 y (α(A)−2)p2dy 1/p2
≤ | | · | |
(cid:16)ZB(0,δ2)c (cid:17) (cid:16)ZB1/2 (cid:17) (2.19)
Cδ2pp22−d 1 p12
≤ · (α(A) 2)p +d
2
h − i
d
C δα(A)−, bychoosing p .
2
≤ · → 2 α(A)
−
−
(cid:0) (cid:1)
Thismeansw = w(x) isα(A) Ho¨ldercontinuousin B . Thisand (2.15)implythat
1/4
−
u C(d, u ).
k kCα(A)−(B1/4) ≤ k kL2(B)
9
ThisprovesCase(I) ofthetheoremwhen n = 0.
We point out that this estimate is almost optimal since the special solution J (x) = x α(A)
0
| |
hasonlyα(A)Ho¨ldercontinuityinB .
1/2
Remark 2.1. Let us pay attention to a special case when A = 1, β = 0, d = 3. According to
theresultsabove,we knowthattheweaksolutionof
1
∆u u = 0 (2.20)
− x 2
| |
inB is Ho¨ldercontinuouswith exponent √5 1 0.618,whichisthegoldenratio.
1/2 2− ≈
−
Next we prove Case (I) of the theorem(cid:0)when(cid:1)n = 1. In this case, we first claim w
∈
C1 B and
1/4
u(y)
(cid:0) (cid:1) ∂ w(x) = ∂ Γ(x y) dy, i = 1,2,...,d. (2.21)
i i − y 2
ZB1/2 | |
Heregoes theproof.
Since nΓsatisfies thefollowingestimate
∇
nΓ(x y) C x y 2 d n, n = 1,2,... (2.22)
− −
|∇ − | ≤ | − |
whereC = C(d,n), thefollowingfunction
u(y)
ξ(x) = ∂ Γ(x y) dy (2.23)
i − y 2
ZB1/2 | |
iswell defined by theHo¨lderinequality. Thereason is:
d
∂ Γ(x y) Lp(B ), 1 p < ;
i 1/2
− ∈ ≤ d 1
− (2.24)
u(y) d
| | C y α(A) 2 Lq(B ), 1 q < .
y 2 ≤ | | − ∈ 1/2 ≤ 2 α(A)
| | −
Thus
α(A) 1 1 1
1 − < + 2 (2.25)
− d p q ≤
and weget
∂ Γ(x y)|u(y)| L(1+d−α(αA(A)−)+11)−(B ). (2.26)
i − y 2 ∈ 1/2
| |
Therefore ξ is well defined. By usual approximation argument, one can prove easily that
∂ w(x) = ξ(x). Thisprovestheclaim.
i
Similarly as in Case(I), n = 0, x ,x B , define z = 1(x +x ), δ = x x . We
∀ 1 2 ∈ 1/4 2 1 2 | 1 − 2|
10
havetheinequalityforthegradient ofw:
1
∂ w(x ) ∂ w(x ) C ∂ Γ(x y) ∂ Γ(x y) dy
| i 1 − i 2 | ≤ | i 1 − − i 2 − | y 2 α(A)
ZB1/2 | | −
1
C ∂ Γ(x y) dy
≤ | i 1 − | y 2 α(A)
ZB1/2TB(z,δ) | | −
1 (2.27)
+C ∂ Γ(x y) dy
| i 2 − | y 2 α(A)
ZB1/2TB(z,δ) | | −
1
+C ∂ Γ(x y) ∂ Γ(x y) dy
| i 1 − − i 2 − | y 2 α(A)
ZB1/2−B(z,δ) | | −
C(I +I +I ).
1 2 3
≡
As toI , ford < p < d , byHo¨lderinequality
1 1 2 α(A)
−
I1 C x1 y (1p−1d−)1p1dy 1−1/p1 y (α(A)−2)p1dy 1/p1
≤ | − | · | |
(cid:16)ZB(x1,32δ) (cid:17) (cid:16)ZB1/2 (cid:17)
Cδp1p−1d 1 p11 2p1 −d −1+1/p1 (2.28)
≤ · (α(A) 2)p +d · p 1
1 1
h − i h − i
d
C δ(α(A) 1)−, by choosing p .
− 1
≤ · → 2 α(A)
−
−
(cid:0) (cid:1)
Likewise,wehavethatI satisfiesthesameestimate. AsforI ,bymeanvalueinequality,there
2 3
existsan xˆ liesbetween x andx , and for1 < p < d :
1 2 2 2 α(A)
−
1
I Cδ 2Γ(xˆ y) dy
3
≤ |∇ − | y 2 α(A)
ZB1/2−B(z,δ) | | −
Cδ y −pd2p−21dy 1−1/p2 y (α(A)−2)p2dy 1/p2
≤ | | · | |
(cid:16)ZB(0,δ2)c (cid:17) (cid:16)ZB1/2 (cid:17) (2.29)
1 1
Cδ1−pd2 p2
≤ · (α(A) 2)p +d
2
h − i
d
C δ(α(A) 1)−, bychoosing p .
− 2
≤ · → 2 α(A)
−
−
(cid:0) (cid:1)
Thismeans w is(α(A) 1) Ho¨ldercontinuousinB andw C1,(α(A) 1) B . More-
1/4 − − 1/4
∇ − − ∈
over
(cid:0) (cid:1)
u C(d, u ). (2.30)
k∇ kC(α(A)−1)− B ≤ k kL2(B)
1/4
ThisshowsCase(I) ofthetheoremwhen n(cid:0)= 1(cid:1)holds.
Now we proveCase(I) ofthe theorem when n > 1. First by induction,it is easy to see the