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Interpolation inequalities for weak solutions of nonlinear parabolic systems
Journal of Inequalities and Applications volume 2011, Article number: 42 (2011)
Abstract
The authors investigate differentiability of the solutions of nonlinear parabolic systems of order 2 m in divergence form of the following type
The achieved results are inspired by the paper of Marino and Maugeri 2008, and the methods there applied.
This note can be viewed as a continuation of the study of regularity properties for solutions of systems started in Ragusa 2002, continued in Ragusa 2003 and Floridia and Ragusa 2012 and also as a generalization of the paper by Capanato and Cannarsa 1981, where regularity properties of the solutions of nonlinear elliptic systems with quadratic growth are reached.
Mathematics Subject Classification (2000)
Primary 35K41, 35K55. Secondary 35B65, 35B45, 35D10
1 Introduction
The study of regularity for solutions of partial differential equations and systems has received considerable attention over the last thirty years. On the other hand, little is known concerning parabolic systems in divergence form of order 2m with quadratic growth and the corresponding analytic properties of solutions. To such classes of systems, our attention is devoted.
This note is a natural continuation of the study, carried out in the last decade and a half, of embedding results of Gagliardo-Nirenberg type from which we deduce local differentiability theorems, making use of interpolation theory in Besov spaces (see e.g. [1–6] and [7]).
In this respect, we mention at first the note [8] where the author proves that, let Ω ⊂ ℝn an open set, 0 < T < ∞ and Q = Ω × (-T, 0), ρ > 0 and , if
is a solution of a second order nonlinear parabolic system of variational type and under the assumptions that the coefficients aα(x, Du) have quadratic growth is obtained that
for every , ∀θ ∈ (0, 1) and for each cube B(2σ) ⊂⊂ Ω.
In the same paper, Fattorusso stressed that it is not possible to improve this result in such a way to achieve, for each solution u to the above system, the differentiability
if, preliminarily, is not ensured the regularity
for every a ∈ (0, T), and for every B(2σ) ⊂⊂ Ω,
The technique used in [8] allows the author to achieve, instead of (1.3), the condition
for every a ∈ (0, T), ∀B(σ) ⊂⊂ Ω and every , which is not enough to ensure that is true (1.2)
In [9], under the same assumptions of the previous result [8], the differentiability result (1.2) is proved, for u satisfying (1.1).
Key of this note is the use of interpolation theorems of Gagliardo-Nirenberg type.
The use of interpolation theory, made in [9] and in [1] with montonicity assumption and quadratic growth, as illustrated in [10], has recently allowed Fattorusso and Marino to obtain differentiability also for weak solutions of nonlinear parabolic systems of second order having nonlinearity 1 < q < 2 (see for details [11]).
Inspired by the note mentioned above by Marino and Maugeri, in the present note, the authors extend differentiability properties to the case of parabolic systems of order 2m. More precisely, let Ω be an open subset of ℝn, n > 2, and 0 < T < ∞, aim of this note is to study, in the cylinder Q = Ω × (-T, 0), the problem of interior local differentiability for solutions
of the nonlinear parabolic systems of order 2m of variational type
Using the above explained idea is proved the following local differentiability with respect to the spatial derivatives
Let us also mention the considerable note by [1] where the authors prove that a solution u of nonlinear parabolic systems of order 2 with natural growth and coefficients uniformly monotone in D u belongs to
Results similar to those obtained by Marino and Maugeri in [9], with stronger assumptions, are obtained by Naumann in [12] and by Naumann and Wolf in [13]. Let us also bear in mind the study made by Campanato in [14] on parabolic systems in divergence form.
We want to finish this historical overview, concerning interior differentiability of weak solutions, recalling the recent note [4] where similar results are achieved for elliptic systems of order 2m.
2 Useful assumptions and results
Let Ω be an bounded open set in ℝn, n > 2, x = (x1, x2, ..., x n ) denotes a generic point therein, 0 < T < ∞ and Q the cylinder Ω × (-T, 0), let N be a positive integer. In Q, we consider the following parabolic metric
Let us set k a positive integer greater than 1, (·|·) k and ||·|| k respectively the scalar product and the norm in ℝk. If there is no ambiguity, we omit the index k.
Let k be a nonnegative integer and λ ∈]0, 1]. We denote by the subspace of of functions that satisfy a Hölder condition of exponent λ, together with all their derivatives Dαu, |α| ≤ k. If , then we set
where
The space is a Banach space, provided with the norm
Definition 2.1 (see e.g. [15, 16]). Let Ω be an bounded open set in ℝn, let k and j be two positive integers, k ≥ j. If p ∈ [1, +∞ [and , so we set
and denote respectively by Hk, p(Ω, ℝN) and the spaces obtained as closure of and regarding the norm ||u||k, p,Ω. The spaces Hk, p(Ω, ℝN) and are known in literature as Sobolev Spaces.
We remark that H0,p(Ω, ℝN) = Lp(Ω, ℝN), 1 ≤ p < +∞.
If p = 2, then we shall simply write Hk(Ω, ℝN), , |u|j,Ω, ||u||k,Ω.
Let Ω be an bounded open set in ℝn, let us set ϑ ∈ (0, 1), p ∈ [1, +∞ [.
Definition 2.2. We say that a function u defined in Ω having values in ℝN belongs to Hϑ, p(Ω, ℝN) if u ∈ Lp(Ω, ℝN) and is finite
Definition 2.3. If k is a nonnegative integer, we mean for Hk+ϑ, p(Ω, ℝN) the subspace of Hk, p(Ω, ℝN) of functions u ∈ Hk,p(Ω, ℝN) such that
We stress that Hk+ϑ, p(Ω, ℝN) is a Banach space equipped with the following norm
If p = 2, then we shall simply write Hk+ϑ(Ω, ℝN) and ||u||k+ϑ,Ω.
Let k a positive integer, p ∈ [1, +∞ [, ϑ ∈ (0, 1), in the following, we will consider the spaces
and
We say a function u ∈ L2(-T, 0, Hm(Ω, ℝN) ∩ Cm-1,λ(Q, ℝN), N positive integer and 0 < λ < 1, weak solution in Q to the nonlinear parabolic system of order 2m
if
Let us now state some properties useful in the sequel.
Let τ ∈]0, 1[, ρ and a two positive numbers and h ∈ ℝ\{0}, where |h| < (1 - τ)ρ.
If u is a function from B(ρ) × (-a, 0) in ℝN and X = (x, t) ∈ B(τρ) × (-a, 0), we set
where {ei}i=1,2,...,nis the canonic basis of ℝn.
Let us now state the following results, proved in [17, 18] and [19], useful to achieve the main result of the note.
Theorem 2.1. If u ∈ Lp(-a, 0, Lp(B(2ρ), ℝN)), a, ρ > 0, 1 < p < +∞, N is a positive integer and exists M > 0 such that
then u ∈ Lp(-a, 0, H1,p(B(ρ), ℝN)) and
Theorem 2.2. Let u ∈ H1,p(B(ρ), ℝN) for a, ρ > 0, 1 ≤ p < +∞ and N be a positive integer. Then, for every τ ∈ (0, 1) and every h ∈ ℝ, |h| < (1 - τ)ρ, we have
Theorem 2.3 (see [18, 20]). Let N be a positive integer and Ω a cube of ℝn. If
with m ≥ 2, m integer, 1 < r < ∞, s ≥ 0, s integer, 0 < λ < 1, s < m - 1, then, for each integer j with s +λ < j < m, there exists two constants c1 and c2 (depending on Ω, m, r, s, λ, j) such that
where , .
Theorem 2.4 (see [9]). Let N be a positive integer and Ω a cube of ℝn. If
with m ≥ 1, m integer, 0 < θ < 1, 1 < r < ∞, s ≥ 0, s integer, 0 < λ < 1, s < m, then, for each integer j with , it results
and there exists a constant c (depending on Ω, m, θ, r, s, λ, j, n, δ) such that
where , , 1[with (1 - δ)(s + λ) + δ(m + θ) noninteger.
3 Interior differentiability of the solutions
Let us set m, N positive integers, α = (α1, ..., α n ) a multi-index and |α| = α1 + · · · + α n the order of α. We denote by the Cartesian product
and , pα ∈ ℝN, the generic point of . If , we set p = (p', p") where , , and
We consider, as usual,
Let us consider the following differential nonlinear variational parabolic system of order 2m :
where aα(X, p) = aα(X, p', p") are functions of in ℝN, satisfying the following conditions:
(3.2) for every α : |α| < m and every , the function X → aα(X, p), defined in Q having values in ℝN, is measurable in X;
(3.3) for every α : |α| < m and every X ∈ Q, the function p → aα(X, p), defined in having values in ℝN, is continuous in p;
(3.4) for every α : |α| < m and every (X, p) ∈ Λ, such that ||p'|| ≤ K, we have
where fα ∈ L2(Q);
(3.5) for every α : |α| = m, the function aα(X, p', p"), defined in having values in ℝN, are of class C1 in and, for every with ||p'|| ≤ K, we have
(3.6) ∃ ν = ν(K) > 0 such that:
for every ξ = (ξα) ∈ R" and for every , with ||p'|| ≤ K. If the coefficients aα satisfy condition (3.6) we say that the system (3.1) is strictly elliptic in Ω.
Theorem 3.1. If u ∈ L2(-T, 0, Hm(Ω, ℝN)) ∩ Cm-1,λ(Q, ℝN), 0 < λ < 1, is a weak solution of the system (3.1) and if the assumptions (3.2)-(3.6) hold, then ∀B(3σ) = B(x0, 3σ) ⊂⊂ Ω, ∀a, b ∈ (0, T), a < b, it results
and the following estimate holds
where K = sup Q ||D' u|| and .
Proof Let us observe that, using Theorem 2.III in [21], for every 0 < ϑ < 1 and , we have
and
Hence, we remark that , then, it results, for a. e. t ∈ (-b*, 0),
Then, from Theorem 2.4 with , 1 - λ < θ < 1, for , and for a.e. t ∈ (-b*, 0):
and there exists a constant c = c(θ, λ, σ, m, n) such that
where .
The choice ensures that for a. e. t ∈ (-b*, 0) we have
and
where .
Then we have, for a. e. t ∈ (-b*, 0), the following inclusion between Sobolev spaces
then, using (3.9), written with , and (3.10)-(3.12), we have
then it follows the requested inequality (3.8). ■
Theorem 3.2 (main result). If u ∈ L2(-T, 0, Hm(Ω, ℝN)) ∩ Cm-1,λ(Q, ℝN), 0< λ < 1, is a weak solution of the system (3.1) and if the assumptions (3.2)-(3.6) hold, then ∀B(3σ) = B(x0, 3σ) ⊂⊂ Ω, ∀a, b ∈ (0, T), a < b it results
and the following estimate holds
where K = sup Q ||D' u|| and .
Proof Let us fix B(3σ) = B(x0, 3σ) ⊂⊂ Ω, a, b ∈ (0, T) with a < b and h ∈ ℝ such that , set , and let a real function satisfying the following properties 0 ≤ ψ ≤ 1 in ℝn, ψ = 1 in B(σ), ψ = 0 in ℝn\B(2σ), in ℝn.
Let us also define the function ρ μ (t), for , μ integer, the following real function
Moreover set {g s (t)} the sequence of symmetric regularizing functions such that
Let i be a positive integer, i ≤ n, and h a real number such that . For every and for every , let us define the following "test function"
Substituting in (2.2) the above defined function φ, we have
For every α : |α| = m and a. e. X = (x, t) ∈ Q, we have
where, if b = b(X, p), for simplicity of notation, we set
Then, equality (3.18) becomes
Taking into account, for α : |α| = m, that
where
we obtain
For s → +∞, using ellipticity condition (3.6), symmetry hypothesis, convolution property of g s and that
we have
where
We observe that, for every ε > 0, we have
The term B can be estimated, for every ε > 0, as follows
Let us consider the term C, for every ε > 0, we have
To estimate the term D, we firstly observe that
then, using Theorem 2.2, we obtain
Finally, using (3.4) condition, the term E can be expressed as follows
Then, from (3.19) estimating the terms A, B, C, D, and E, for every ε > 0, we have
We observe that the function
is continuous in the origin, then ∃h0(ν, K, U, λ, σ, m, n), , such that for every |h| < h0, we have
For each integer i = 1, ..., n, for and every h such that |h| < h0(< 1), it follows
Let us focus our attention on the last term, taking into account that from (3.12), for a. e. t ∈ (-b*, 0), we have
then using Hölder and Young inequalities, for every α such that |α| < m, for every ε > 0, it follows
Furthermore, for every α such that |α| < m, from Theorem 2.2 for every h ∈ ℝ with |h| < h0 and for every ε > 0, we have
the last inequality follows, as before, applying Theorem 2.2 for p = 2.
Let us now choose , it ensures
Multiplying each term for and integrating respect to and applying (3.13), we achieve
Taking into consideration the last inequality and the properties of the function ψ, from (3.31) we deduce
From which, passing the limit μ → ∞, we get
Let us now estimate the last term in (3.32). Using Hölder inequality, applying Theorem 2.2 (for p = 4, instead of B(σ) and ) and formula (3.13), for every |h| < h0, it follows
Integrating in (-b*, 0), from (3.32), it follows
If , for every i = 1, 2, ..., n we easily obtain
It is then proved, for every and every i ∈ {1, 2, ..., n}, that
applying Theorem 2.1, it follows
and
Finally we have to prove that u ∈ H1(-a, 0, L2(B(σ), ℝN)) and inequality (3.15).
From inequality (3.8), we have
then we have
Moreover, bearing in mind that, for |α| < m, aα(X, p) satisfies (3.4), and for |α| = m, aα(X, p) satisfies (3.5), we have
Recalling the definition of weak solution, for every , proceeding as in [22], we have
and, bearing in mind (3.37), we obtain that
From (3.4), (3.5) and (3.38), we get
The last inequality and (3.34) allows us to conclude the proof. ■
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Floridia, G., Ragusa, M.A. Interpolation inequalities for weak solutions of nonlinear parabolic systems. J Inequal Appl 2011, 42 (2011). https://doi.org/10.1186/1029-242X-2011-42
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DOI: https://doi.org/10.1186/1029-242X-2011-42