Let X0,X1,…,Xq(q<N) be real vector fields, which are left invariant on homogeneous group G, provided that X0 is homogeneous of degree two and X1,…,Xq are homogeneous of degree one. We consider the following nondivergence degenerate operator with drift L=∑i,j=1qaij(x)XiXj+a0(x)X0, where the coefficients aij(x), a0(x) belonging to vanishing mean oscillation space are bounded measurable functions. Furthermore, aij(x) satisfies the uniform ellipticity condition on Rq and a0(x)≠0. We obtain the local weighted Sobolev–Morrey estimates by applying the boundedness of commutators and interpolation inequalities on weighted Morrey spaces.
We consider a boundary value problem in a bounded domain involving a degenerate operator of the form L ( u ) = − div ( a ( x ) ∇ u ) and a suitable nonlinearity f. The function a vanishes on smooth 1-codimensional submanifolds of Ω where it is not allowed to be C 2 . By using weighted Sobolev spaces we are still able to find existence of solutions which vanish, in the trace sense, on the set where a vanishes.
AbstractActinobacteria are among the most prolific sources of medically and agriculturally important compounds, derived from their biosynthetic gene clusters (BGCs) for specialized (secondary) pathways of metabolism. Genomics witnesses that the majority of actinobacterial BGCs are silent, most likely due to their low or zero transcription. Much effort is put into the search for approaches towards activation of silent BGCs, as this is believed to revitalize the discovery of novel natural products. We hypothesized that the global transcriptional factor AdpA, due to its highly degenerate operator sequence, could be used to upregulate the expression of silent BGCs. Using Streptomyces cyanogenus S136 as a test case, we showed that plasmids expressing either full-length adpA or its DNA-binding domain led to significant changes in the metabolome. These were evident as changes in the accumulation of colored compounds, bioactivity, as well as the emergence of a new pattern of secondary metabolites as revealed by HPLC-ESI-mass spectrometry. We further focused on the most abundant secondary metabolite and identified it as the polyene antibiotic lucensomycin. Finally, we uncovered the entire gene cluster for lucensomycin biosynthesis (lcm), that remained elusive for five decades until now, and outlined an evidence-based scenario for its adpA-mediated activation.
The issues of well-posedness of linear inverse coefficient problems for multi-term equations in Banach spaces with fractional Riemann – Liouville derivatives and with bounded operators at them are considered. Well-posedness criteria are obtained both for the equation resolved with respect to the highest fractional derivative, and in the case of a degenerate operator at the highest derivative in the equation. Two essentially different cases are investigated in the degenerate problem: when the fractional part of the order of the second-oldest derivative is equal to or different from the fractional part of the order of the highest fractional derivative. Abstract results are applied in the study of inverse problems for partial differential equations with polynomials from a self-adjoint elliptic differential operator with respect to spatial variables and with Riemann – Liouville derivatives in time.
An initial-boundary value problem for a system of third-order partial differential equations is considered. Equations and systems of equations with the highest mixed third derivative describe heat exchange in the soil complicated by the movement of soil moisture, quasi-stationary processes in a two-component semiconductor plasma, etc. The system is reduced to a differential equation with a degenerate operator at the highest derivative with respect to the distinguished variable in a Banach space. This operator has the property of having 0 as a normal eigenvalue, which makes it possible to split the original equations into an equation in subspaces. The conditions are obtained under which a unique solution to the problem exists; the analytical formula is found.
We are concerned with the study of a bifurcation problem driven by a degenerate operator of Baouendi–Grushin type. Due to its degenerate structure, this differential operator has a mixed regime. Studying the combined effects generated by the absorption and the reaction terms, we establish the bifurcation behavior in two cases. First, if the absorption nonlinearity is dominating, then the problem admits solutions only for high perturbations of the reaction. In the case when the reaction dominates the absorption term, we prove that the problem admits nontrivial solutions for all the values of the parameter. The analysis developed in this paper is associated with patterns describing transonic flow restricted to subsonic regions.
Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups
