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Remarks on Hölder continuity for solutions of the p-Laplace evolution equations
J. Math. Anal. Appl. 382 (2011) 785–791
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Remarks on Hölder continuity for solutions of the p-Laplace evolution equations Masashi Mizuno Mathematical Institute, Tohoku University, Aoba, Sendai, Japan
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 19 November 2010 Available online 30 April 2011 Submitted by P. Sacks
We study the interior Hölder regularity problem for the gradient of solutions of the pLaplace evolution equations with the external forces. Misawa gave some conditions for the Hölder continuity of the gradient of solutions. We show Hölder estimates of the solutions with weaker condition as for Misawa. © 2011 Elsevier Inc. All rights reserved.
Keywords: Hölder continuity Degenerate parabolic equation p-Laplace evolution equations with the external forces
1. Introduction We consider the following p-Laplace evolution equations:
∂t u − div |∇ u | p −2 ∇ u = div f , t > 0, x ∈ Rn , u (0, x) = u 0 (x), x ∈ Rn ,
(1.1)
n n n n where p > n2n +2 is a constant, u : (0, ∞) × R → R is unknown, f : (0, ∞) × R → R and u 0 : R → R are given external and initial data. It is well known that a classical solution of (1.1) does not generally exist. Hence we introduce the notion of weak solutions. p
Definition 1.1. For u 0 ∈ L 2 (Rn ) and f ∈ L p−1 ((0, ∞) × Rn ), we call u a weak solution of (1.1) if there exists T > 0 such that (1) u ∈ L ∞ (0, T ; L 2 (Rn )) with ∇ u ∈ L p (0, T ; L p (Rn )); and (2) u satisfies (1.1) in the sense of distribution, namely, for all
t u (t )ϕ (t ) dx −
Rn
t u ∂t ϕ dτ dx +
0 Rn
|∇ u |
p −2
0 Rn
ϕ ∈ C 1 (0, T ; C 01 (Rn )) and for almost all 0 < t < T , t
∇ u · ∇ ϕ dτ dx =
u 0 ϕ (0) dx − Rn
f · ∇ ϕ dτ dx. 0 Rn
For the existence of a weak solution is shown by Browder [1], Ladyženskaja, Solonnikov and Ural’ceva [7] (cf. Ôtani [10]). In this paper, we study the Hölder continuity of ∇ u, particularly, we show a relationship between the Hölder continuity of ∇ u and the regularity of the external force f .
E-mail address: [email protected]. 0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.04.076
©
2011 Elsevier Inc. All rights reserved.
786
M. Mizuno / J. Math. Anal. Appl. 382 (2011) 785–791
The Hölder continuity of ∇ u was firstly shown by DiBenedetto and Friedman [4,5] and Wiegner [11] when f ≡ 0. Misawa [8] showed the Hölder continuity of ∇ u if f is locally Hölder continuous with respect to t and x. In this paper, we give a weaker condition of the external force f for the Hölder continuity of ∇ u than the condition given by Misawa. To state the main theorem, we introduce some notation. For t 0 ∈ R and R > 0, we write an open interval I R (t 0 ) := (t 0 − R 2 , t 0 ). For x0 ∈ Rn and R > 0, we denote the n-dimensional open ball with radius R and center x0 by B R (x0 ). We define a parabolic cylinder Q R (t 0 , x0 ) by Q R (t 0 , x0 ) := I R (t 0 ) × B R (t 0 ). For an integrable function f on a measurable set A, we denote an integral mean ( f ) A by
( f ) A :=
1
| A|
f dx. A
Theorem 1.2. Assume that for some constant K > 0 and γ > n + 2, the external force f satisfies
f − f (t ) B
R (x0 )
p p−1 dt dx K R γ
(1.2)
Q R (t 0 ,x0 )
for all (t 0 , x0 ) ∈ (0, ∞) × Rn and 0 < R < 1 satisfying Q R (t 0 , x0 ) ⊂ (0, ∞) × Rn . Let u be a weak solution of (1.1). Then ∇ u is Hölder continuous with exponent γ > 0 depending only on n, p, γ . Furthermore, for all ε > 0, there exists a constant C > 0 depending only on n, p, γ , K , ε and ∇ u L ∞ ((0,∞)×Rn ) such that
γ ∇ u (t , x) − ∇ u (s, y ) C |t − s| 2 + |x − y |γ , for all (t , x), (s, y ) ∈ (ε , ∞) × Rn . Remark 1.3. If f is Hölder continuous in (0, ∞) × Rn , then the assumption (1.2) holds. Furthermore, if ∇ f ∈ L r (0, ∞; L q (Rn )) with 2r + nq < 1, then the assumption (1.2) holds. Thus, we need not the assumption that f is Hölder continuous with respect to t. Remark 1.4. The assumption that ∇ u belongs to L ∞ ((0, ∞) × Rn ) can be replaced by the assumption that for some M > 0,
|∇ u | p dx dt M R n+2−σ
(1.3)
Q R (t 0 ,x0 )
for sufficiently small σ > 1 and for Q R (t 0 , x0 ) ⊂ (0, ∞) × Rn , namely Morrey regularity for |∇ u | p . If the external force f is in the space of functions of bounded mean oscillation, then ∇ u ∈ L r ((0, ∞) × Rn ) for all r > p by Misawa [9] and the assumption (1.3) holds. Since the assumption (1.2) implies the boundedness of mean oscillation of f , the boundedness of ∇ u is not an essential assumption. Remark 1.5. We only treat the case p > 2 in this paper since we may obtain the Hölder continuity of ∇ u for the case 2n < p < 2 by almost same argument. Especially, we may show the Morrey regularity of |∇ u | p for the case n2n n+2 +2 < p.
Furthermore, even for the case 1 < p n2n +2 , we may also show the local Hölder continuity of ∇ u by assuming the Morrey regularity of |∇ u | p (cf. Choe [2]). At the end of this section, we introduce further notation. For a parabolic cylinder Q R (t 0 , x0 ), we define a parabolic boundary ∂ p Q R (t 0 , x0 ) by
∂ p Q R (t 0 , x0 ) :=
t 0 − R 2 × B R (x0 ) ∪ I R (t 0 ) × ∂ B R (x0 ) .
For (t , x), (s, y ) ∈ R × Rn , we write a parabolic distance dist p ((t , x), (s, y )) by
1
dist p (t , x), (s, y ) := max |t − s| 2 , |x − y | . For A , B ⊂ R × Rn , we write a parabolic distance dist p ( A , B ) by
dist p ( A , B ) :=
inf
z∈ A , z ∈ B
dist p z, z .
We denote a constant depending on α , β, . . . by C (α , β, . . .). The same letter C will be used to denote difference constants. We use subscript numbers if we consider the relation between the constants.
M. Mizuno / J. Math. Anal. Appl. 382 (2011) 785–791
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2. Proof of Theorem 1.2 From Remarks 1.4 and 1.5, we may assume without loss of generality that p > 2 and the boundedness of the gradient of the solution. By the scaling argument, we only consider (t 0 , x0 ) ∈ (1, ∞) × Rn and we will show the Hölder continuity at (t 0 , x0 ). We abbreviate the center of open balls and parabolic cylinders. For R < 1, we consider the following reference equation:
∂t v − div |∇ v | p −2 ∇ v = 0, (t , x) ∈ Q R , v = u, (t , x) ∈ ∂ p Q R .
(2.1)
For the existence of a solution of (1.2), we refer to Ladyženskaja, Solonnikov and Ural’ceva [7, Theorem 6.7 on p. 466]. Lemma 2.1. There exists a constant C > 0 depending only on n, p such that
( v − u )2 dx|t =0 + BR
|∇ v − ∇ u | p dt dx C QR
p f − f (t ) p−1 dt dx. B R
QR
Proof. Subtract (1.1) from (2.1), multiply ( v − u ) and integrate in Q R . Then we obtain
1
∂t ( v − u )2 dt dx +
2 QR
=
|∇ v | p −2 ∇ v − |∇ u | p −2 ∇ u · (∇ v − ∇ u ) dt dx
QR
f − f (t ) B · (∇ v − ∇ u ) dt dx.
R
QR
We utilize algebraic inequalities
|p| p −2 p − |q| p −2 q · (p − q) C 0 |p − q| p ,
which hold for all p, q ∈ Rn , where C 0 is a positive constant depending only on n, p (cf. DiBenedetto [3, Lemma 4.4 on p. 13]). By the Hölder inequality and the Young inequality, we have
1
∂t ( v − u )2 dt dx + C 0 (n, p )
2 QR
|∇ v − ∇ u | p dt dx QR
f − f (t ) B · (∇ v − ∇ u ) dt dx
R
QR
1− 1p
1p p p f − f (t ) p−1 dt dx |∇ v − ∇ u | dt dx B R
QR
C 0 (n, p )
QR
|∇ v − ∇ u | p dt dx + C 1 (n, p )
2 QR
p f − f (t ) p−1 dt dx. B R
2
QR
Remark 2.2. When 1 < p < 2, we utilize the following algebraic inequalities:
|p| p −2 p − |q| p −2 q · (p − q) C 0 |p − q|2 |p| + |q| ,
which hold for all p, q ∈ Rn , where C 0 is a positive constant depending only on n, p. The following lemma is given by DiBenedetto [3]. Lemma 2.3. (See DiBenedetto [3, Theorem 5.1 on p. 238].) There exists a constant C > 0 depending only on n, p such that
sup |∇ v | C Q
R 2
1
|Q R|
p
12
|∇ v | dt dx
.
QR
Using Lemma 2.3, we obtain the following lemma:
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M. Mizuno / J. Math. Anal. Appl. 382 (2011) 785–791
Lemma 2.4. There exists a constant C > 0 depending only on n, p such that
1
p
1
sup |∇ v | C K 2 R 2 (γ −n−2) + ∇ u L2∞ ((0,∞)×Rn ) . Q
(2.2)
R 2
In particular, |∇ v | is bounded on Q R . 2
Proof. By Lemma 2.1 and the assumption (1.2), we have
|∇ v | p dt dx C ( p ) |∇ v − ∇ u | p dt dx + |∇ u | p dt dx
QR
QR
QR
p−p 1 p C ( p) f − f (t ) B dt dx + | Q R |∇ u L ∞ ((0,∞)×Rn ) R
QR
p C ( p ) K R γ + | Q R |∇ u L ∞ ((0,∞)×Rn ) . 2
Using Lemma 2.3, we obtain (2.2).
Remark 2.5. If the assumption (1.3) holds, then we have the following growth estimate of |∇ v |:
1
1
σ
sup |∇ v | C K 2 + M 2 R − 2 . Q
R 2
The next lemma is given by DiBenedetto [3]. Lemma 2.6. (See DiBenedetto [3, Theorem 1.1 on p. 256].) For 0 < δ < 1 and R < 1, there exist constants α0 > 0 and C > 0 depending only on n, p such that p −2 2
osc(∇ v ) C ∇ v L ∞ ( Q Q
R 2
R + R max{1, ∇ v L ∞ ( Q R 1−δ ) } α0 R 1−δ 2
2
)
dist p ( Q R , ∂ p Q 2
R 1−δ 2
.
)
Using Lemma 2.6 we will show the following lemma: Lemma 2.7. For 0 < δ < 1, there exists a constant C > 0 depending only on n, p such that,
osc(∇ v ) 8R δ ∇ v L ∞ ( Q Q
R 2
R 1−δ 2
)
p −2 + C ∇ v L ∞ ( Q R 1−δ ) 1 + max 1, ∇ v L ∞2 ( Q 2
α0 R 1−δ 2
)
R δ α0 .
Proof. Either if R δ 14 , then
osc(∇ v ) 2∇ v L ∞ ( Q R ) 8R δ ∇ v L ∞ ( Q Q
R 2
2
R 1−δ 2
).
Otherwise, if 0 < R δ < 14 , then
dist( Q R , ∂ p Q 2
R 1−δ 2
) = min =
R
1−δ
2
R 1−δ
2
2
2 12 1−δ
R R R − , − 2
1 − Rδ
3 8
2
2
R 1−δ .
Therefore, by Lemma 2.6, we obtain
osc(∇ v ) C ( p , n)∇ v L ∞ ( Q Q
R 2
R 1−δ 2
α0
p −2
2 ) 1 + max 1, ∇ v L ∞ ( Q
R 1−δ 2
)
R δ α0 .
2
We slightly modify an iteration argument, since the monotonicity of the oscillation of ∇ u is not easily obtained.
M. Mizuno / J. Math. Anal. Appl. 382 (2011) 785–791
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Lemma 2.8. (Cf. Giaquinta [6, Lemma 2.1 on p. 86].) Let φ = φ(s) be a non-negative function on [0, ∞). Assume that for some constants R 0 , A 0 , A 1 , B , α , β > 0 with β < α , the function φ satisfies
φ(ρ ) A 0 φ( R ), α ρ φ( R ) + B R β φ(ρ ) A 1 R
for all 0 < ρ R R 0 . Then, there exists a constant C > 0 depending only on A 0 , A 1 , α , β such that
β
ρ
φ(ρ ) C
R
φ( R ) + B ρ β .
Proof. Let r < 1 be chosen later and for R R 0 , let
ρ = r R. Then by the assumption we have
φ(r R ) A 1 r α φ( R ) + B R β . γ
We fix β < γ < α and we put r = r0 satisfying A 1 r0α r0 . Then γ
φ(r0 R ) A 1 r0α φ( R ) + B R β r0 φ( R ) + B R β . Therefore, for k ∈ N ∪ {0}, we have k (k+1)γ j (γ −β) kβ φ r0k+1 R r0 φ( R ) + Br0 R β r0 . j =0
Since k
j (γ −β)
r0
∞
j =0
j (γ −β)
r0
j =0
β
=
r0 β
γ,
r0 − r0
we obtain β kγ kβ B R φ r0k R r0 φ( R ) + r0 β γ r0 − r0
ρ > 0, there exists k0 ∈ N ∪ {0} such that r0k0 +1 R ρ r0k0 R. Therefore, we obtain β
k kγ kβ B R φ(ρ ) A 0 φ r00 R A 0 r0 φ( R ) + r0 β γ r0 − r0 γ β
B Rβ −γ ρ −β ρ 2 φ( R ) + r0 A 0 r0 γ . β R R r0 − r0
for k ∈ N ∪ {0}. For
Proof of Theorem 1.2. Fix 0 < ρ < R < 1. Then
∇ u − (∇ u ) Q p dt dx C ( p ) ∇ v − (∇ v ) Q p dt dx . |∇ u − ∇ v | p dt dx + ρ ρ
Qρ
Qρ
First, we estimate
Qρ
Qρ
(2.3)
Qρ
|∇ v − (∇ v ) Q ρ | p dt dx. Either if 0 < ρ R2 , then by Lemma 2.7
∇ v − (∇ v ) Q p dt dx ρ
osc(∇ v ) p dt dx Qρ p
C (n, p )∇ v L ∞ ( Q
R 1−δ 2
)R
p
+ C (n, p )∇ v L ∞ ( Q
n +2 + p δ
R 1−δ 2
p α0
p −2
2 ) 1 + max 1, ∇ v L ∞ ( Q
R 1−δ 2
)
R n +2 + p δ α 0 .
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M. Mizuno / J. Math. Anal. Appl. 382 (2011) 785–791
Otherwise, if
R 2
ρ R, then
n+2+ p ∇ v − (∇ v ) Q p dt dx 2n+2+ p ρ ∇ v − (∇ v ) Q p dt dx ρ ρ
R
Qρ
Qρ
2n+2+ p
n+2+ p
∇ v − (∇ v ) Q p dt dx. R
ρ R
QR
Since
∇ v − (∇ v ) Q p dt dx C (n, p ) R
QR
|∇ v − ∇ u | dt dx + C (n, p ) QR
we obtain
p
∇ u − (∇ u ) Q p dt dx, R
QR
n+2+ p ∇ v − (∇ v ) Q p dt dx C (n, p ) ρ ∇ u − (∇ u ) Q p dt dx + ∇ v p∞ ρ R L (Q R
Qρ
QR
p
+ C (n, p )∇ v L ∞ ( Q
p α0
p −2
2 ) 1 + max 1, ∇ v L ∞ ( Q
R 1−δ 2
)
)R
n +2 + p δ
R n +2 + p δ α 0
|∇ u − ∇ v | p dt dx.
+ C (n, p ) Next, we estimate
R 1−δ 2
R 1−δ 2
QR
|∇ v − ∇ u | p dt dx. By Lemma 2.1, we have QR
|∇ u − ∇ v | p dt dx C (n, p ) K R γ . QR
Therefore, by (2.3), we obtain
n+2+ p ∇ u − (∇ u ) Q p dt dx C (n, p ) ρ ∇ u − (∇ u ) Q p dt dx ρ R R
Qρ
QR
+
p C (n, p )∇ v L ∞ ( Q p
+ C (n, p )∇ v L ∞ ( Q
R 1−δ 2
)R
n +2 + p δ
p −2
2 ) 1 + max 1, ∇ v L ∞ ( Q R 1−δ ) R 1−δ 2
+ C (n, p ) K R γ . Since
γ > n + 2, we can apply Lemma 2.8 for φ(ρ ) = ∇ u − (∇ u ) Q p dt dx C ρ n+2+ p γ ρ
p α0
R n +2 + p δ α 0
2
(2.4)
|∇ u − (∇ u ) Q ρ | dt dx and we obtain p
Qρ
Qρ
for some
γ > 0. Therefore ∇ u is Hölder continuous with exponent γ > 0. 2
Remark 2.9. If the assumption (1.3) holds, then we can modify the estimate (2.4) by
n+2+ p ∇ u − (∇ u ) Q p dt dx C ρ ∇ u − (∇ u ) Q p dt dx ρ R R
Qρ
QR σ
+ C R n+2+ p δ− 2 (1−δ) p + C R n+2+ p δ α0 − 2 p (1−δ)(1+ σ
σ > 0, we obtain n+2+ p ∇ u − (∇ u ) Q p dt dx C ρ ∇ u − (∇ u ) Q p dt dx + C R n+2+ p γ ρ R
Therefore for sufficiently small
R
Qρ
for some
p −2 p α0 )
QR
γ > 0 and we can apply Lemma 2.8.
+ C K Rγ .
M. Mizuno / J. Math. Anal. Appl. 382 (2011) 785–791
791
Acknowledgments The author would like to express his deepest gratitude to Professor Takayoshi Ogawa for his valuable comments and encouragement. He also thanks the referees for recommending various improvements in exposition. This work is supported by the JSPS Research Fellowships for Young Scientists and the JSPS Grant-in-Aid for the JSPS fellows #21-1281.
References [1] F.E. Browder, Nonlinear parabolic boundary value problems of arbitrary order, Bull. Amer. Math. Soc. 69 (1963) 858–861. [2] H.-J. Choe, Hölder regularity for the gradient of solutions of certain singular parabolic systems, Comm. Partial Differential Equations 16 (1991) 1709– 1732. [3] E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. [4] E. DiBenedetto, A. Friedman, Regularity of solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math. 349 (1984) 83–128. [5] E. DiBenedetto, A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985) 1–22. [6] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. of Math. Stud., vol. 105, Princeton University Press, Princeton, NJ, 1983. [7] O.A. Ladyženskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr., vol. 23, American Mathematical Society, Providence, RI, 1967, translated from Russian by S. Smith. [8] M. Misawa, Local Hölder regularity of gradients for evolutional p-Laplacian systems, Ann. Mat. Pura Appl. (4) 181 (2002) 389–405. [9] M. Misawa, L q estimates of gradients for evolutional p-Laplacian systems, J. Differential Equations 219 (2005) 390–420. [10] M. Ôtani, L ∞ -energy method and its applications, in: Nonlinear Partial Differential Equations and Their Applications, in: GAKUTO Internat. Ser. Math. ¯ Sci. Appl., vol. 20, Gakkotosho, Tokyo, 2004, pp. 505–516. [11] M. Wiegner, On C α -regularity of the gradient of solutions of degenerate parabolic systems, Ann. Mat. Pura Appl. (4) 145 (1986) 385–405.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Remarks on Hölder continuity for solutions of the p-Laplace evolution equations Masashi Mizuno Mathematical Institute, Tohoku University, Aoba, Sendai, Japan
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 19 November 2010 Available online 30 April 2011 Submitted by P. Sacks
We study the interior Hölder regularity problem for the gradient of solutions of the pLaplace evolution equations with the external forces. Misawa gave some conditions for the Hölder continuity of the gradient of solutions. We show Hölder estimates of the solutions with weaker condition as for Misawa. © 2011 Elsevier Inc. All rights reserved.
Keywords: Hölder continuity Degenerate parabolic equation p-Laplace evolution equations with the external forces
1. Introduction We consider the following p-Laplace evolution equations:
∂t u − div |∇ u | p −2 ∇ u = div f , t > 0, x ∈ Rn , u (0, x) = u 0 (x), x ∈ Rn ,
(1.1)
n n n n where p > n2n +2 is a constant, u : (0, ∞) × R → R is unknown, f : (0, ∞) × R → R and u 0 : R → R are given external and initial data. It is well known that a classical solution of (1.1) does not generally exist. Hence we introduce the notion of weak solutions. p
Definition 1.1. For u 0 ∈ L 2 (Rn ) and f ∈ L p−1 ((0, ∞) × Rn ), we call u a weak solution of (1.1) if there exists T > 0 such that (1) u ∈ L ∞ (0, T ; L 2 (Rn )) with ∇ u ∈ L p (0, T ; L p (Rn )); and (2) u satisfies (1.1) in the sense of distribution, namely, for all
t u (t )ϕ (t ) dx −
Rn
t u ∂t ϕ dτ dx +
0 Rn
|∇ u |
p −2
0 Rn
ϕ ∈ C 1 (0, T ; C 01 (Rn )) and for almost all 0 < t < T , t
∇ u · ∇ ϕ dτ dx =
u 0 ϕ (0) dx − Rn
f · ∇ ϕ dτ dx. 0 Rn
For the existence of a weak solution is shown by Browder [1], Ladyženskaja, Solonnikov and Ural’ceva [7] (cf. Ôtani [10]). In this paper, we study the Hölder continuity of ∇ u, particularly, we show a relationship between the Hölder continuity of ∇ u and the regularity of the external force f .
E-mail address: [email protected]. 0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.04.076
©
2011 Elsevier Inc. All rights reserved.
786
M. Mizuno / J. Math. Anal. Appl. 382 (2011) 785–791
The Hölder continuity of ∇ u was firstly shown by DiBenedetto and Friedman [4,5] and Wiegner [11] when f ≡ 0. Misawa [8] showed the Hölder continuity of ∇ u if f is locally Hölder continuous with respect to t and x. In this paper, we give a weaker condition of the external force f for the Hölder continuity of ∇ u than the condition given by Misawa. To state the main theorem, we introduce some notation. For t 0 ∈ R and R > 0, we write an open interval I R (t 0 ) := (t 0 − R 2 , t 0 ). For x0 ∈ Rn and R > 0, we denote the n-dimensional open ball with radius R and center x0 by B R (x0 ). We define a parabolic cylinder Q R (t 0 , x0 ) by Q R (t 0 , x0 ) := I R (t 0 ) × B R (t 0 ). For an integrable function f on a measurable set A, we denote an integral mean ( f ) A by
( f ) A :=
1
| A|
f dx. A
Theorem 1.2. Assume that for some constant K > 0 and γ > n + 2, the external force f satisfies
f − f (t ) B
R (x0 )
p p−1 dt dx K R γ
(1.2)
Q R (t 0 ,x0 )
for all (t 0 , x0 ) ∈ (0, ∞) × Rn and 0 < R < 1 satisfying Q R (t 0 , x0 ) ⊂ (0, ∞) × Rn . Let u be a weak solution of (1.1). Then ∇ u is Hölder continuous with exponent γ > 0 depending only on n, p, γ . Furthermore, for all ε > 0, there exists a constant C > 0 depending only on n, p, γ , K , ε and ∇ u L ∞ ((0,∞)×Rn ) such that
γ ∇ u (t , x) − ∇ u (s, y ) C |t − s| 2 + |x − y |γ , for all (t , x), (s, y ) ∈ (ε , ∞) × Rn . Remark 1.3. If f is Hölder continuous in (0, ∞) × Rn , then the assumption (1.2) holds. Furthermore, if ∇ f ∈ L r (0, ∞; L q (Rn )) with 2r + nq < 1, then the assumption (1.2) holds. Thus, we need not the assumption that f is Hölder continuous with respect to t. Remark 1.4. The assumption that ∇ u belongs to L ∞ ((0, ∞) × Rn ) can be replaced by the assumption that for some M > 0,
|∇ u | p dx dt M R n+2−σ
(1.3)
Q R (t 0 ,x0 )
for sufficiently small σ > 1 and for Q R (t 0 , x0 ) ⊂ (0, ∞) × Rn , namely Morrey regularity for |∇ u | p . If the external force f is in the space of functions of bounded mean oscillation, then ∇ u ∈ L r ((0, ∞) × Rn ) for all r > p by Misawa [9] and the assumption (1.3) holds. Since the assumption (1.2) implies the boundedness of mean oscillation of f , the boundedness of ∇ u is not an essential assumption. Remark 1.5. We only treat the case p > 2 in this paper since we may obtain the Hölder continuity of ∇ u for the case 2n < p < 2 by almost same argument. Especially, we may show the Morrey regularity of |∇ u | p for the case n2n n+2 +2 < p.
Furthermore, even for the case 1 < p n2n +2 , we may also show the local Hölder continuity of ∇ u by assuming the Morrey regularity of |∇ u | p (cf. Choe [2]). At the end of this section, we introduce further notation. For a parabolic cylinder Q R (t 0 , x0 ), we define a parabolic boundary ∂ p Q R (t 0 , x0 ) by
∂ p Q R (t 0 , x0 ) :=
t 0 − R 2 × B R (x0 ) ∪ I R (t 0 ) × ∂ B R (x0 ) .
For (t , x), (s, y ) ∈ R × Rn , we write a parabolic distance dist p ((t , x), (s, y )) by
1
dist p (t , x), (s, y ) := max |t − s| 2 , |x − y | . For A , B ⊂ R × Rn , we write a parabolic distance dist p ( A , B ) by
dist p ( A , B ) :=
inf
z∈ A , z ∈ B
dist p z, z .
We denote a constant depending on α , β, . . . by C (α , β, . . .). The same letter C will be used to denote difference constants. We use subscript numbers if we consider the relation between the constants.
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2. Proof of Theorem 1.2 From Remarks 1.4 and 1.5, we may assume without loss of generality that p > 2 and the boundedness of the gradient of the solution. By the scaling argument, we only consider (t 0 , x0 ) ∈ (1, ∞) × Rn and we will show the Hölder continuity at (t 0 , x0 ). We abbreviate the center of open balls and parabolic cylinders. For R < 1, we consider the following reference equation:
∂t v − div |∇ v | p −2 ∇ v = 0, (t , x) ∈ Q R , v = u, (t , x) ∈ ∂ p Q R .
(2.1)
For the existence of a solution of (1.2), we refer to Ladyženskaja, Solonnikov and Ural’ceva [7, Theorem 6.7 on p. 466]. Lemma 2.1. There exists a constant C > 0 depending only on n, p such that
( v − u )2 dx|t =0 + BR
|∇ v − ∇ u | p dt dx C QR
p f − f (t ) p−1 dt dx. B R
QR
Proof. Subtract (1.1) from (2.1), multiply ( v − u ) and integrate in Q R . Then we obtain
1
∂t ( v − u )2 dt dx +
2 QR
=
|∇ v | p −2 ∇ v − |∇ u | p −2 ∇ u · (∇ v − ∇ u ) dt dx
QR
f − f (t ) B · (∇ v − ∇ u ) dt dx.
R
QR
We utilize algebraic inequalities
|p| p −2 p − |q| p −2 q · (p − q) C 0 |p − q| p ,
which hold for all p, q ∈ Rn , where C 0 is a positive constant depending only on n, p (cf. DiBenedetto [3, Lemma 4.4 on p. 13]). By the Hölder inequality and the Young inequality, we have
1
∂t ( v − u )2 dt dx + C 0 (n, p )
2 QR
|∇ v − ∇ u | p dt dx QR
f − f (t ) B · (∇ v − ∇ u ) dt dx
R
QR
1− 1p
1p p p f − f (t ) p−1 dt dx |∇ v − ∇ u | dt dx B R
QR
C 0 (n, p )
QR
|∇ v − ∇ u | p dt dx + C 1 (n, p )
2 QR
p f − f (t ) p−1 dt dx. B R
2
QR
Remark 2.2. When 1 < p < 2, we utilize the following algebraic inequalities:
|p| p −2 p − |q| p −2 q · (p − q) C 0 |p − q|2 |p| + |q| ,
which hold for all p, q ∈ Rn , where C 0 is a positive constant depending only on n, p. The following lemma is given by DiBenedetto [3]. Lemma 2.3. (See DiBenedetto [3, Theorem 5.1 on p. 238].) There exists a constant C > 0 depending only on n, p such that
sup |∇ v | C Q
R 2
1
|Q R|
p
12
|∇ v | dt dx
.
QR
Using Lemma 2.3, we obtain the following lemma:
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M. Mizuno / J. Math. Anal. Appl. 382 (2011) 785–791
Lemma 2.4. There exists a constant C > 0 depending only on n, p such that
1
p
1
sup |∇ v | C K 2 R 2 (γ −n−2) + ∇ u L2∞ ((0,∞)×Rn ) . Q
(2.2)
R 2
In particular, |∇ v | is bounded on Q R . 2
Proof. By Lemma 2.1 and the assumption (1.2), we have
|∇ v | p dt dx C ( p ) |∇ v − ∇ u | p dt dx + |∇ u | p dt dx
QR
QR
QR
p−p 1 p C ( p) f − f (t ) B dt dx + | Q R |∇ u L ∞ ((0,∞)×Rn ) R
QR
p C ( p ) K R γ + | Q R |∇ u L ∞ ((0,∞)×Rn ) . 2
Using Lemma 2.3, we obtain (2.2).
Remark 2.5. If the assumption (1.3) holds, then we have the following growth estimate of |∇ v |:
1
1
σ
sup |∇ v | C K 2 + M 2 R − 2 . Q
R 2
The next lemma is given by DiBenedetto [3]. Lemma 2.6. (See DiBenedetto [3, Theorem 1.1 on p. 256].) For 0 < δ < 1 and R < 1, there exist constants α0 > 0 and C > 0 depending only on n, p such that p −2 2
osc(∇ v ) C ∇ v L ∞ ( Q Q
R 2
R + R max{1, ∇ v L ∞ ( Q R 1−δ ) } α0 R 1−δ 2
2
)
dist p ( Q R , ∂ p Q 2
R 1−δ 2
.
)
Using Lemma 2.6 we will show the following lemma: Lemma 2.7. For 0 < δ < 1, there exists a constant C > 0 depending only on n, p such that,
osc(∇ v ) 8R δ ∇ v L ∞ ( Q Q
R 2
R 1−δ 2
)
p −2 + C ∇ v L ∞ ( Q R 1−δ ) 1 + max 1, ∇ v L ∞2 ( Q 2
α0 R 1−δ 2
)
R δ α0 .
Proof. Either if R δ 14 , then
osc(∇ v ) 2∇ v L ∞ ( Q R ) 8R δ ∇ v L ∞ ( Q Q
R 2
2
R 1−δ 2
).
Otherwise, if 0 < R δ < 14 , then
dist( Q R , ∂ p Q 2
R 1−δ 2
) = min =
R
1−δ
2
R 1−δ
2
2
2 12 1−δ
R R R − , − 2
1 − Rδ
3 8
2
2
R 1−δ .
Therefore, by Lemma 2.6, we obtain
osc(∇ v ) C ( p , n)∇ v L ∞ ( Q Q
R 2
R 1−δ 2
α0
p −2
2 ) 1 + max 1, ∇ v L ∞ ( Q
R 1−δ 2
)
R δ α0 .
2
We slightly modify an iteration argument, since the monotonicity of the oscillation of ∇ u is not easily obtained.
M. Mizuno / J. Math. Anal. Appl. 382 (2011) 785–791
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Lemma 2.8. (Cf. Giaquinta [6, Lemma 2.1 on p. 86].) Let φ = φ(s) be a non-negative function on [0, ∞). Assume that for some constants R 0 , A 0 , A 1 , B , α , β > 0 with β < α , the function φ satisfies
φ(ρ ) A 0 φ( R ), α ρ φ( R ) + B R β φ(ρ ) A 1 R
for all 0 < ρ R R 0 . Then, there exists a constant C > 0 depending only on A 0 , A 1 , α , β such that
β
ρ
φ(ρ ) C
R
φ( R ) + B ρ β .
Proof. Let r < 1 be chosen later and for R R 0 , let
ρ = r R. Then by the assumption we have
φ(r R ) A 1 r α φ( R ) + B R β . γ
We fix β < γ < α and we put r = r0 satisfying A 1 r0α r0 . Then γ
φ(r0 R ) A 1 r0α φ( R ) + B R β r0 φ( R ) + B R β . Therefore, for k ∈ N ∪ {0}, we have k (k+1)γ j (γ −β) kβ φ r0k+1 R r0 φ( R ) + Br0 R β r0 . j =0
Since k
j (γ −β)
r0
∞
j =0
j (γ −β)
r0
j =0
β
=
r0 β
γ,
r0 − r0
we obtain β kγ kβ B R φ r0k R r0 φ( R ) + r0 β γ r0 − r0
ρ > 0, there exists k0 ∈ N ∪ {0} such that r0k0 +1 R ρ r0k0 R. Therefore, we obtain β
k kγ kβ B R φ(ρ ) A 0 φ r00 R A 0 r0 φ( R ) + r0 β γ r0 − r0 γ β
B Rβ −γ ρ −β ρ 2 φ( R ) + r0 A 0 r0 γ . β R R r0 − r0
for k ∈ N ∪ {0}. For
Proof of Theorem 1.2. Fix 0 < ρ < R < 1. Then
∇ u − (∇ u ) Q p dt dx C ( p ) ∇ v − (∇ v ) Q p dt dx . |∇ u − ∇ v | p dt dx + ρ ρ
Qρ
Qρ
First, we estimate
Qρ
Qρ
(2.3)
Qρ
|∇ v − (∇ v ) Q ρ | p dt dx. Either if 0 < ρ R2 , then by Lemma 2.7
∇ v − (∇ v ) Q p dt dx ρ
osc(∇ v ) p dt dx Qρ p
C (n, p )∇ v L ∞ ( Q
R 1−δ 2
)R
p
+ C (n, p )∇ v L ∞ ( Q
n +2 + p δ
R 1−δ 2
p α0
p −2
2 ) 1 + max 1, ∇ v L ∞ ( Q
R 1−δ 2
)
R n +2 + p δ α 0 .
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M. Mizuno / J. Math. Anal. Appl. 382 (2011) 785–791
Otherwise, if
R 2
ρ R, then
n+2+ p ∇ v − (∇ v ) Q p dt dx 2n+2+ p ρ ∇ v − (∇ v ) Q p dt dx ρ ρ
R
Qρ
Qρ
2n+2+ p
n+2+ p
∇ v − (∇ v ) Q p dt dx. R
ρ R
QR
Since
∇ v − (∇ v ) Q p dt dx C (n, p ) R
QR
|∇ v − ∇ u | dt dx + C (n, p ) QR
we obtain
p
∇ u − (∇ u ) Q p dt dx, R
QR
n+2+ p ∇ v − (∇ v ) Q p dt dx C (n, p ) ρ ∇ u − (∇ u ) Q p dt dx + ∇ v p∞ ρ R L (Q R
Qρ
QR
p
+ C (n, p )∇ v L ∞ ( Q
p α0
p −2
2 ) 1 + max 1, ∇ v L ∞ ( Q
R 1−δ 2
)
)R
n +2 + p δ
R n +2 + p δ α 0
|∇ u − ∇ v | p dt dx.
+ C (n, p ) Next, we estimate
R 1−δ 2
R 1−δ 2
QR
|∇ v − ∇ u | p dt dx. By Lemma 2.1, we have QR
|∇ u − ∇ v | p dt dx C (n, p ) K R γ . QR
Therefore, by (2.3), we obtain
n+2+ p ∇ u − (∇ u ) Q p dt dx C (n, p ) ρ ∇ u − (∇ u ) Q p dt dx ρ R R
Qρ
QR
+
p C (n, p )∇ v L ∞ ( Q p
+ C (n, p )∇ v L ∞ ( Q
R 1−δ 2
)R
n +2 + p δ
p −2
2 ) 1 + max 1, ∇ v L ∞ ( Q R 1−δ ) R 1−δ 2
+ C (n, p ) K R γ . Since
γ > n + 2, we can apply Lemma 2.8 for φ(ρ ) = ∇ u − (∇ u ) Q p dt dx C ρ n+2+ p γ ρ
p α0
R n +2 + p δ α 0
2
(2.4)
|∇ u − (∇ u ) Q ρ | dt dx and we obtain p
Qρ
Qρ
for some
γ > 0. Therefore ∇ u is Hölder continuous with exponent γ > 0. 2
Remark 2.9. If the assumption (1.3) holds, then we can modify the estimate (2.4) by
n+2+ p ∇ u − (∇ u ) Q p dt dx C ρ ∇ u − (∇ u ) Q p dt dx ρ R R
Qρ
QR σ
+ C R n+2+ p δ− 2 (1−δ) p + C R n+2+ p δ α0 − 2 p (1−δ)(1+ σ
σ > 0, we obtain n+2+ p ∇ u − (∇ u ) Q p dt dx C ρ ∇ u − (∇ u ) Q p dt dx + C R n+2+ p γ ρ R
Therefore for sufficiently small
R
Qρ
for some
p −2 p α0 )
QR
γ > 0 and we can apply Lemma 2.8.
+ C K Rγ .
M. Mizuno / J. Math. Anal. Appl. 382 (2011) 785–791
791
Acknowledgments The author would like to express his deepest gratitude to Professor Takayoshi Ogawa for his valuable comments and encouragement. He also thanks the referees for recommending various improvements in exposition. This work is supported by the JSPS Research Fellowships for Young Scientists and the JSPS Grant-in-Aid for the JSPS fellows #21-1281.
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