Nonvariational basic parabolic systems of second order

Sergio Campanato

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1991)

  • Volume: 2, Issue: 2, page 129-136
  • ISSN: 1120-6330

Abstract

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Ω is a bounded open set of R n , of class C 2 and T > 0 . In the cylinder Q = Ω × 0 , T we consider non variational basic operator a ⁢ H ⁢ u - ∂ ⁡ u / ∂ ⁡ t where a ⁢ ξ is a vector in R N , N ≥ 1 , which is continuous in ξ and satisfies the condition (A). It is shown that ∀ f ∈ L 2 ⁢ Q the Cauchy-Dirichlet problem u ∈ W 0 2 , 1 ⁢ Q , a ⁢ H ⁢ u - ∂ ⁡ u / ∂ ⁡ t = f in Q , has a unique solution. It is further shown that if u ∈ W 0 2 , 1 ⁢ Q is a solution of the basic system a ⁢ H ⁢ u - ∂ ⁡ u / ∂ ⁡ t = 0 in Q , then H ⁢ u and ∂ ⁡ u / ∂ ⁡ t belong to H l ⁢ o ⁢ c 1 ⁢ Q . From this the Hölder continuity in Q of the vectors u and D ⁢ u are deduced respectively when n ≤ 4 and n = 2 .

How to cite

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Campanato, Sergio. "Nonvariational basic parabolic systems of second order." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 2.2 (1991): 129-136. <http://eudml.org/doc/244088>.

@article{Campanato1991,
abstract = {\( \Omega \) is a bounded open set of \( \mathbb\{R\}^\{n\} \), of class \( C^\{2\} \) and \( T>0 \). In the cylinder \( Q = \Omega \times (0, T) \) we consider non variational basic operator \( a(H(u)) - \partial u / \partial t \) where \( a(\xi) \) is a vector in \( \mathbb\{R\}^\{N\} \), \( N \ge 1 \), which is continuous in \( \xi \) and satisfies the condition (A). It is shown that \( \forall f \in L^\{2\} (Q) \) the Cauchy-Dirichlet problem \( u \in W\_\{0\}^\{2,1\} (Q) \), \( a(H(u)) - \partial u / \partial t = f \) in \( Q \), has a unique solution. It is further shown that if \( u \in W\_\{0\}^\{2,1\} (Q) \) is a solution of the basic system \( a(H(u)) - \partial u / \partial t = 0 \) in \( Q \), then \( H(u) \) and \( \partial u / \partial t \) belong to \( H^\{1\}\_\{loc\} (Q) \). From this the Hölder continuity in \( Q \) of the vectors \( u \) and \( D u \) are deduced respectively when \( n \le 4 \) and \( n = 2 \).},
author = {Campanato, Sergio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Nonlinear non variational systems; (A) condition; Existence theorem; Cauchy-Dirichlet problem; Hölder continuity of solutions; adjacency to linear operator; smoothness of solutions},
language = {eng},
month = {6},
number = {2},
pages = {129-136},
publisher = {Accademia Nazionale dei Lincei},
title = {Nonvariational basic parabolic systems of second order},
url = {http://eudml.org/doc/244088},
volume = {2},
year = {1991},
}

TY - JOUR
AU - Campanato, Sergio
TI - Nonvariational basic parabolic systems of second order
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1991/6//
PB - Accademia Nazionale dei Lincei
VL - 2
IS - 2
SP - 129
EP - 136
AB - \( \Omega \) is a bounded open set of \( \mathbb{R}^{n} \), of class \( C^{2} \) and \( T>0 \). In the cylinder \( Q = \Omega \times (0, T) \) we consider non variational basic operator \( a(H(u)) - \partial u / \partial t \) where \( a(\xi) \) is a vector in \( \mathbb{R}^{N} \), \( N \ge 1 \), which is continuous in \( \xi \) and satisfies the condition (A). It is shown that \( \forall f \in L^{2} (Q) \) the Cauchy-Dirichlet problem \( u \in W_{0}^{2,1} (Q) \), \( a(H(u)) - \partial u / \partial t = f \) in \( Q \), has a unique solution. It is further shown that if \( u \in W_{0}^{2,1} (Q) \) is a solution of the basic system \( a(H(u)) - \partial u / \partial t = 0 \) in \( Q \), then \( H(u) \) and \( \partial u / \partial t \) belong to \( H^{1}_{loc} (Q) \). From this the Hölder continuity in \( Q \) of the vectors \( u \) and \( D u \) are deduced respectively when \( n \le 4 \) and \( n = 2 \).
LA - eng
KW - Nonlinear non variational systems; (A) condition; Existence theorem; Cauchy-Dirichlet problem; Hölder continuity of solutions; adjacency to linear operator; smoothness of solutions
UR - http://eudml.org/doc/244088
ER -

References

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  1. CAMPANATO, S., L 2 , λ theory for non linear non variational differential systems. Rendiconti di Matematica di Roma, to appear. Zbl0777.35028
  2. CAMPANATO, S., Non variational differential systems. A condition for local existence and uniqueness. Proceedings of the Caccioppoli Conference, to appear. Zbl0796.35052MR1306303
  3. CAMPANATO, S., A Cordes type condition for nonlinear non variational systems. Rend. Acc. Naz. delle Scienze detta dei XL, vol. XDI, 1989. Zbl0702.35084
  4. CAMPANATO, S., Sistemi ellittici in forma divergenza. Regolarità all'interno. Quaderni della Scuola Normale Superiore di Pisa, 1980. Zbl0453.35026MR668196
  5. CAMPANATO, S., Non variational basic elliptic systems of second order. Rendiconti del Seminario Matematico e Fisico di Milano, to appear. Zbl0786.35028MR1229486DOI10.1007/BF02925081
  6. CAMPANATO, S., Sul problema di Cauchy-Dirichlet per equazioni paraboliche del secondo ordine, non variazionali, a coefficienti discontinui. Rendiconti Sem. Matem. Padova, vol. XLI, 1968. Zbl0202.37602MR252857
  7. CANNARSA, P., Second order non variational parabolic systems. Boll. U.M.I., Serie V, vol XVHI, C.N.1., 1981. Zbl0473.35043

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