A Carleman Function and the Cauchy Problem for the Laplace by Yarmukhamedov Sh.

By Yarmukhamedov Sh.

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The following gives a bounded slope condition having this effect; we return to the generic notation. 4 Theorem Let X be an open subset of Rm . Suppose that there exist positive constants c, d such that x ∈ X, v ∈ Γ(x), (α, β) ∈ NGP (x, v) =⇒ |α| ≤ {c |v| + d} |β| . 5. PROXIMAL CRITERIA FOR PSEUDO-LIPSCHITZNESS 55 Let any x∗ ∈ X and v∗ ∈ Γ(x∗ ) be given, together with ε > 0 such that B(x∗ , ε) ⊂ X. Set 1 . δ∗ := min ε, 3(1 + 2c) Then for any N ≥ 1, the multifunction Γ is pseudo-Lipschitz of radius RN := (c |v∗ | + d)N near (x∗ , v∗ ) as follows: for any x, x ∈ B(x∗ , δ∗ ) we have Γ(x ) ∩ B(v∗ , RN ) ⊂ Γ(x) + kN x − x B, where kN := (c |v∗ | + d)(1 + 2cN ).

3) with the following data: f (x, v) := IG (x, v), Y := {x2 } × B(v1 , k |x2 − x1 |). We take the base point (x1 , v1 ), and a tolerance ε˜ small enough in a sense to be specified presently. If G ∩ Y is nonempty, the existence of the required point v2 follows. Therefore we suppose that G ∩ Y is empty, and we proceed to get a contradiction. Since (under the emptiness assumption) minY f = +∞, the Mean Value Inequality asserts that for any r > 0 we may write r < ζ, x2 − x1 + ψ, v − v1 + ε˜ for all v ∈ B(v1 , k |x2 − x1 |), where (ζ, ψ) belongs to ∂P f (z, w), and where (z, w) is a point within distance ε˜ of the set co [Y ∪ {(x1 , v1 )}].

2 36 CHAPTER 2. 18) (via the second term in the minimum). There remains Case 3. From |fi (t)| = (1 − λi (t)η)RN (t) we derive |λi (t)fi (t)| ≥ (λ0 − η)RN (t). With this in mind we put λ = 1 and f ∈ F (t, xi (t)) ∩ B(0, r0 ) in the Weierstrass condition and deduce pi (t), λi (t)fi (t) − f + εi (1 + 4N 2 ) ≥ φ((λ0 − η)RN (t)). 8). End of the proof We now complete the proof of the theorem. We showed above that there exist p and γ (not both 0) satisfying the transversality and Euler conditions, and the Weierstrass condition of radius (λ0 − 2η)RN (t).

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