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Boyan Sirakov

Publications and source records attributed to Boyan Sirakov.

At least 19 recordsLinked to original sources
arXiv2026

A priori estimates and exact solvability for non-coercive stochastic control equations

We establish, for the first time, explicit a priori and regularity estimates for solutions of the Dirichlet problem for Hamilton-Jacobi-Bellman operators from stochastic control, whose principal half-eigenvalues have opposite signs. In addition, if the negative eigenvalue is not too negative, the problem can have exactly two, one or zero solutions, depending on the valuation function. This is a novel exact multiplicity result for fully nonlinear equations, which also yields a generalization of the Ambrosetti-Prodi theorem to such equations.

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arXiv2025

Keller-Osserman and Harnack type results for nonlinear elliptic PDE with unbounded ingredients

We show that the classical Keller-Osserman theorem on the solvability of the equation $\mathcal{L}[u] = f(u)$ is valid when $\mathcal{L}$ is a general operator in divergence form with unbounded coefficients in the natural regime of local integrability. This has been open up to now, earlier results concerned operators with locally bounded ingredients. We also settle an open question from \cite{SS21} about the validity of the strong maximum principle for supersolutions of $\mathcal{L}[u] = f(u)$ under the optimal integral condition of V\'azquez. More generally, we obtain a Harnack inequality for positive solutions of this equation, which extends a result by V. Julin.

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arXiv2025

Hopf-Oleinik lemma for elliptic equations in double divergence form

We establish, for the first time, a Zaremba-Hopf-Oleinik type boundary point lemma for uniformly elliptic partial differential equations in double divergence form, also known as stationary Fokker-Planck-Kolmogorov equations. As an application, we derive sharp two-sided estimates for the Green's function associated with second-order elliptic equations in non-divergence form in $C^{1,\alpha}$ domains.

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arXiv2024

Elliptic regularity estimates with optimized constants and applications

We revisit the classical theory of linear second-order uniformly elliptic equations in divergence form whose solutions have H\"older continuous gradients, and prove versions of the generalized maximum principle, the $C^{1,\alpha}$-estimate, the Hopf-Oleinik lemma, the boundary weak Harnack inequality and the differential Harnack inequality, in which the constant is optimized with respect to the norms of the coefficients of the operator and the size of the domain. Our estimates are complemented by counterexamples which show their optimality. We also give applications to the Landis conjecture and spectral estimates.

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arXiv2022

Boundary weak Harnack estimates and regularity for elliptic PDE in divergence form

We obtain a global extension of the classical weak Harnack inequality which extends and quantifies the Hopf-Oleinik boundary-point lemma, for uniformly elliptic equations in divergence form. Among the consequences is a boundary gradient estimate, due to Krylov and well-studied for non-divergence form equations, but completely novel in the divergence framework. Another consequence is a new more general version of the Hopf-Oleinik lemma.

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arXiv2022

Uniform a priori estimates for positive solutions of the Lane-Emden system in the plane

We prove that positive solutions of the superlinear Lane-Emden system in a two-dimensional smooth bounded domain are bounded independently of the exponents in the system, provided the exponents are comparable. As a consequence, the energy of the solutions is uniformly bounded. In addition, the boundedness may fail if the exponents are not comparable.

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arXiv2021

Sharp boundary and global regularity for degenerate fully nonlinear elliptic equations

We obtain optimal boundary and global regularity estimates for viscosity solutions of fully nonlinear elliptic equations whose ellipticity degenerates at the critical points of a given solution. We show that any solution is $C^{1,α}$ on the boundary of the domain, for an optimal and explicit $α$ given only in terms of the regularity of the boundary datum and the elliptic degeneracy degree, no matter how possibly low is the interior regularity for that class of equations. We also obtain sharp global estimates. Our findings are new even for model equations, involving only a degenerate Laplacian; all previous results of global nature give $C^{1,α}$ regularity only for some small $α>0$.

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arXiv2021

The Vázquez maximum principle and the Landis conjecture for elliptic PDE with unbounded coefficients

We develop a new, unified approach to the following two classical questions on elliptic PDE: the strong maximum principle for equations with non-Lipschitz nonlinearities, and the at most exponential decay of solutions in the whole space or exterior domains. Our results apply to divergence and nondivergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a (weak) Harnack inequality with optimal dependence of the constants in the lower-order terms of the equation and the size of the domain, which we establish.

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arXiv2020

Global integrability and boundary estimates for uniformly elliptic PDE in divergence form

We show that two classically known properties of positive supersolutions of uniformly elliptic PDEs, the boundary point principle (Hopf lemma) and global integrability, can be quantified with respect to each other. We obtain an extension to the boundary of the de Giorgi-Moser weak Harnack inequality, optimal with respect to the norms involved, for equations in divergence form.

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arXiv2020

A Liouville-type theorem for the Lane-Emden equation in a half-space

We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solution which is monotone in the normal direction. As a consequence, this problem does not admit any positive classical solution which is bounded on finite strips. This question has a long history and our result solves a long-standing open problem. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. The result extends to general convex nonlinearities.

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arXiv2020

Liouville-type theorems for unbounded solutions of elliptic equations in half-spaces

We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solutions which grow at most like the distance to the boundary to a power given by the natural scaling exponent of the equation; in other words, we rule out {\it type~I grow-up} solutions. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. Instrumental in the proof are local pointwise bounds for the logarithmic gradient of the solution and its normal derivative, which we also establish.

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arXiv2019

A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth

We consider fully nonlinear uniformly elliptic cooperative systems with quadratic growth in the gradient, such as $$ -F_i(x, u_i, Du_i, D^2 u_i)- \langle M_i(x)D u_i, D u_i \rangle =λc_{i1}(x) u_1 + \cdots + λc_{in}(x) u_n +h_i(x), $$ for $i=1,\cdots,n$, in a bounded $C^{1,1}$ domain $Ω\subset \mathbb{R}^N$ with Dirichlet boundary conditions; here $n\geq 1$, $λ\in\mathbb{R}$, $c_{ij},\, h_i \in L^\infty(Ω)$, $c_{ij}\geq 0$, $M_i$ satisfies $0<μ_1 I\leq M_i\leq μ_2 I$, and $F_i$ is an uniformly elliptic Isaacs operator. We obtain uniform a priori bounds for systems, under a weak coupling hypothesis that seems to be optimal. As an application, we also establish existence and multiplicity results for these systems, including a branch of solutions which is new even in the scalar case.

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arXiv2019

A new method of proving a priori bounds for superlinear elliptic PDE

We describe a new method of proving a priori bounds for positive supersolutions and solutions of superlinear elliptic PDE, based on global weak Harnack inequalities and a quantitative Hopf lemma. Novel results based on the method include: (i) equations without a boundary condition on the whole boundary; (ii) equations with nonlinearities which do not have precise growth at infinity; (iii) systems of inequalities with opposite sign.

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arXiv2018

A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as $$ -F(x,u,Du,D^2u) =λc(x)u+\langle M(x)D u, D u \rangle +h(x) $$ in a bounded domain with a Dirichlet boundary condition, here $λ\in\mathbb{R}$, $c,\, h \in L^p(Ω)$, $p>n\geq 1$, $c\gneqq 0$ and the matrix $M$ satisfies $0<μ_1 I\leq M\leq μ_2 I$. Recently this problem was studied in the "coercive" case $λc\le0$, where uniqueness of solutions can be expected, and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle. We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vázquez type strong maximum principle for our kind of equations.

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arXiv2017

Results of Ambrosetti-Prodi type for non-selfadjoint elliptic operators

The well-known Ambrosetti-Prodi theorem considers perturbations of the Dirichlet Laplacian by a nonlinear function whose derivative jumps over the principal eigenvalue of the operator. Various extensions of this landmark result were obtained for self-adjoint operators, in particular by Berger and Podolak, who gave a geometrical description of the solution set. In this text we show that similar theorems are valid for non self-adjoint operators. In particular, we prove that the semilinear operator is a global fold. As a consequence, we obtain what appears to be the first exact multiplicity result for elliptic equations in non-divergence form. We employ techniques based on the maximum principle.

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arXiv2015

Stationary states of reaction-diffusion and Schrödinger systems with inhomogeneous or controlled diffusion

We obtain classification, solvability and nonexistence theorems for positive stationary states of reaction-diffusion and Schrödinger systems involving a balance between repulsive and attractive terms. This class of systems contains PDE arising in biological models of Lotka-Volterra type, in physical models of Bose-Einstein condensates and in models of chemical reactions. We show, with different proofs, that the results obtained in [ARMA, 213 (2014), 129-169] for models with homogeneous diffusion are valid for general heterogeneous media, and even for controlled inhomogeneous diffusions.

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