Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

Abstract

Àííîòàöèÿ. Ïóñòü D îòêðûòîå ñâÿçíîå ìíîaeåñòâî ñ äîñòàòî÷íî ãëàä-êîé ãðàíèöåé ∂D íà êîìïëåêñíîé ïëîñêîñòè C. Âîçìóùàÿ çàäà÷ó Êîøè äëÿ ñèñòåìû Êîøè-Ðèìàíà ∂u = f â D ñ ãðàíè÷íûìè äàííûìè íà çà-ìêíóòîì ìíîaeåñòâå S ⊂ ∂D, ìû ïîëó÷àåì ñåìåéñòâî ñìåøàííûõ çàäà÷ òèïà Çàðåìáû äëÿ óðàâíåíèÿ Ëàïëàñà, çàâèñÿùåå îò ìàëîãî ïàðàìåòðà ε ∈ (0, 1] â ãðàíè÷íîì óñëîâèè. Íåñìîòðÿ íà òî, ÷òî ñìåøàííûå çàäà-÷è ñîäåðaeàò íåêîýðöèòèâíûå ãðàíè÷íûå óñëîâèÿ íà ∂D \ S, êàaeäàÿ èç íèõ èìååò åäèíñòâåííîå ðåøåíèå â ïîäõîäÿùåì ãèëüáåðòîâîì ïðîñòðàí-

On Initial Boundary Value Problem for Parabolic Differential Operator with Non-coercive Boundary Conditions

“…In general case the condition (11) have no sense for functions u ∈ L 2 (0, T ; H + (Ω)). But we will see below that function u(t) ∈ L 2 (0, T ; H + (Ω)), satisfying (10), is continuous and (11) have a sense.…”

“…However the problem with non-coercive boubdary conditions are also appeared in both theory and applications, see, for instance, pioneer work in this direction [5] and papers [6,7] and [8] for such problems in the Elasticity Theory. Recent results in Fredholm operator equations, induced by boundary value problems for elliptic differential operators with non-coercive boundary conditions (see, for instance, [9][10][11][12]) allows us to apply these one for studying the parabolic problem. Consideration of such problems essentially extends variety of boundary operators, but there is a loss of regularity of the solution (see [13] for elliptic case).…”

Regularization of the Cauchy Problem for Elliptic Operators

“…Thus we have described the closure of the image of the map (2). Description of the image of the map (2) itself is a more difficult task.…”

“…Recently, a new approach was developed, cf. [2] based on the simple observation that the calculus of the solutions to the Cauchy problems foran elliptic equations just amounts to the calculus of a (possibly non-coercive) mixed boundary value problems for an elliptic equations with a parameter.…”

On initial boundary value problem for parabolic differential operator with non-coercive boundary conditions

“…However the problem with non-coercive boubdary conditions are also appeared in both theory and applications, see, for instance, pioneer work in this direction [5] and papers [6], [7] and [8] for such problems in the Elasticity Theory. Recent results in Fredholm operator equations, induced by boundary value problems for elliptic differential operators with non-coercive boundary conditions (see, for instance, [9], [10], [11], [12]) allows us to apply these one for studying the parabolic problem. Consideration of such problems essentially extends variety of boundary operators, but there is a loss of regularity of the solution (see [13] for elliptic case).…”