Sufficient Conditions for First-Order Differential Operators to be Associated with a q -Metamonogenic Operator in a Clif
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Sufficient Conditions for First-Order Differential Operators to be Associated with a q-Metamonogenic Operator in a Clifford Type Algebra Eusebio Ariza García1 · Antonio Di Teodoro2 · María Sapiain3 · Franklin Vargas4
Received: 10 November 2015 / Accepted: 28 June 2016 © Springer-Verlag Berlin Heidelberg 2016
Abstract Consider the initial value problem ∂t u = L(t, x, u, ∂xi u), u(0, x) = ϕ(x),
(0.1)
where t is the time, L is a linear first-order differential operator and ϕ is a generalized q-metamonogenic function. This problem can be solved by applying the method of associated spaces which is constructed by Tutschke (see Solution of initial value problems in classes of generalized analytic functions, Teubner Leipzig and Springer, New York, 1989). In this work, we formulate sufficient conditions
Communicated by Stephan Ruscheweyh.
B
Eusebio Ariza García [email protected] Antonio Di Teodoro [email protected] María Sapiain [email protected] Franklin Vargas [email protected]
1
School of Mathematics, Yachay Tech, Yachay City of Knowledge, Urcuquí, Ecuador
2
Department of mathematics, The Pennsylvania State University, 109 McAllister Building, University Park, PA 16802, USA
3
Escuela de Matemáticas, Universidad Central de Venezuela, Caracas, Venezuela
4
Departamento de Cómputo Científico y Estadística, Universidad Simón Bolívar, Caracas, Venezuela
123
E. Ariza et al.
on the coefficients of the operator L under which this operator is associated to the space of generalized q-metamonogenic functions satisfying a differential equation with anti-q-metamonogenic right-hand side, when q and λ are constant Clifford vectors. We also build a computational algorithm to check the computations in the cases A∗2,2 and A∗3,2 . In conical domains, the initial value problem (0.1) is uniquely solvable for an operator L and for any generalized q-metamonogenic initial function ϕ, provided an interior estimate holds for generalized q-metamonogenic functions satisfying a differential equation with anti-q-metamonogenic right-hand side. The solution is also a generalized q-metamonogenic function for each fixed t. This work generalizes the results given in Di Teodoro and Sapian (Adv. Appl. Clifford Algebras, 25:283–301, 2015) and Van (Differential operator in a Clifford analysis associated to differential equations with anti-monogenic right hand side, IC/2006/134, 2016). Keywords Initial value problem · Associated spaces · Associated Operators · Interior estimates Mathematics Subject Classification 35B45 · 35F10 · 47H10 · 34A12 · 11E88
1 Introduction Consider the initial value problem ∂t u = L(t, x, u, ∂xi u), u(0, x) = ϕ(x),
(1.1) (1.2)
where t is the time, L is a linear (in u and its derivatives) first–order differential operator. The classical Cauchy–Kovalevskaya problem for an evolution equation with a holomorphic right-hand side states that each initial value problem with holomorphic initial data is (uniquely) solvable. On the other hand, the famous Lewy example (see [13]) shows that there are infinitely different
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