On Solvability Conditions for a Certain Conjugation Problem

New Solvability Conditions for Congruence

K. Bibak et al. [arXiv:1503.01806v1 [math.NT], March 5 2015] proved that congruence ax ≡ b (mod n) has a solution x0 with t = gcd(x0, n) if and only if gcd ( a, n t ) = gcd ( b t , n t ) thereby generalizing the result for t = 1 proved by B. Alomair et al. [J. Math. Cryptol. 4 (2010), 121–148] and O. Grošek et al. [ibid. 7 (2013), 217–224]. We show that this generalized result for arbitrary t f...

Necessary and Sufficient Conditions for the Solvability of the L Dirichlet Problem on Lipschitz Domains

We study the homogeneous elliptic systems of order 2l with real constant coefficients on Lipschitz domains in R, n ≥ 4. For any fixed p > 2, we show that a reverse Hölder condition with exponent p is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in L. We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the L ...

Solvability Conditions for a Nonlocal Boundary Value Problem for Linear Functional Differential Equations

The aim of the paper is to find efficient conditions for the unique solvability of the problem u′(t) = `(u)(t) + q(t), u(a) = h(u) + c, where ` : C([a, b];R) → L([a, b];R) and h : C([a, b];R) → R are linear bounded operators, q ∈ L([a, b];R), and c ∈ R.

Necessary and sufficient conditions for the solvability of the Lp Dirichlet problem on Lipschitz domains

We study the homogeneous elliptic systems of order 2 with real constant coefficients on Lipschitz domains in Rn, n ≥ 4. For any fixed p > 2, we show that a reverse Hölder condition with exponent p is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in Lp. We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the L...

Solvability for Parabolic Poincaré Problem