Abstract
We apply the method of integral representations to study the ill-posed Cauchy problem for the heat equa- tion. More precisely we recover a function, satisfying the heat equation in a cylindrical domain, via its values and the values of its normal derivative on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural (anisotropic) spaces (Sobolev and H¨older spaces, etc). Finally, we obtain a uniqueness theorem for the problem and a criterion of its solvability and a Carleman-type formula for its solution
Highlights
The integral representation method is the core of investigations of the ill-posed problems for partial differential equations, see [1,2,3]
Lev Aizenberg [4, 5] noted that the Cauchy problem for the Cauchy–Riemann equations is closely related to the problem of analytic continuation even if its entries are not analytic
He found principal ingredients, leading to the construction of integral formulas for its solution (Carleman formulas): a proper integral formula recovering the function via the data on the whole boundary, the uniqueness theorem and an effective tool, providing the analytic continuation from a domain to a larger one
Summary
Received 28.02.2019, received in revised form 11.03.2019, accepted 20.04.2019 We apply the method of integral representations to study the ill-posed Cauchy problem for the heat equation. He found principal ingredients, leading to the construction of integral formulas for its solution (Carleman formulas): a proper integral formula recovering the function via the data on the whole boundary, the uniqueness theorem and an effective tool, providing the analytic continuation from a domain to a larger one This method was successfully used in the framework of Hilbert space methods to investigate the Cauchy problem for general elliptic systems of partial differential equations, see [6,7,8], and even to elliptic complexes of differential operators, see [9]. To investigate the heat equation we need the anisotropic (parabolic) spaces, see [12, Ch. 1], [14, Ch. 8] For this purpose, let C2s,s(ΩT ) stand for the set of all continuous functions u in ΩT , (α, j) ∈ Zn+ × Z+. Let C2s+k,s,λ,λ/2((Ω ∪ S)T ) stand for the set of anisotropic Holder continuous functions with a power λ over each compact subset of (Ω ∪ S)T together with all partial derivatives ∂xα+β∂tju where |β| k, |α| + 2j 2s. It is known that the standard initial boundary value problem for the heat operator consists of the recovering of the function u over the cylinder
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