Related papers: A degenerate operator in non divergence form

In this paper we consider the fourth order operators A1u := (au")" and A2u := au"" in divergence form and non divergence form, respectively, where a, defined in [0, 1] with values in R+, degenerates in an interior point of the interval.…

We study the generation property for a fourth order operator in divergence or in non divergence form with suitable Neumann boundary conditions. As a consequence we obtain the well posedness for the parabolic equations governed by these…

In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form $$ Au(x)=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}e^{i(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)dyd\xi, $$ where…

We consider five different abstract Cauchy problems where the operator is of fourth order. For each problem, using semigroups techniques and functional calculus, we study the invertibility and the spectral properties of the fourth order…

The aim of this paper is to present some results about generation, sectoriality and gradient estimates both for the semigroup and for the resolvent of suitable realizations of the operators [\Ab u(x)=\gamma xu"(x) + b u'(x),] with constants…

We prove that operators of the form $A=-a(x)^2\Delta^{2}$, with suitable growth conditions on the coefficient $a(x)$, generate analytic semigroups in $L^1(\mathbb{R}^N)$. In particular, we deduce generation results for the operator $A :=-…

It is shown that the theory of real symmetric second-order elliptic operators in divergence form on $\Ri^d$ can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behaviour of the…

In this paper we consider higher order Schr\"odinger operators $$\mathcal L u=Lu+Vu,$$ where $L$ denotes a fourth order operator and $V\geq 0$ a suitable potential. We initiate our analysis by considering the constant coefficients…

We in this paper study the nonexpansive operators equipped with arbitrary metric and investigate the connections between firm nonexpansiveness, cocoerciveness and averagedness. The convergence of the associated fixed-point iterations is…

In a recent paper we presented a general perturbation result for generators of $C_0$-semigroups. The aim of the present paper is to replace, in case the unperturbed semigroup is analytic, the various conditions appearing in this result by…

We develop a theory of "special functions" associated to a certain fourth order differential operator $\mathcal{D}_{\mu,\nu}$ on $\mathbb{R}$ depending on two parameters $\mu,\nu$. For integers $\mu,\nu\geq-1$ with $\mu+\nu\in2\mathbb{N}_0$…

We study geometric properties of varieties associated with invariant subspaces of nilpotent operators. There are reductive algebraic groups acting on these varieties. We give dimensions of orbits of these actions. Moreover, a combinatorial…

We prove that operators of the form $A=-a(x)^2\Delta^{2}$, with $|D a(x)|\leq c a(x)^\frac{1}{2}$, generate analytic semigroups in $L^p(\mathbb{R}^N)$ for $1<p\leq\infty$ and in $C_b(\mathbb{R}^N)$. In particular, we deduce generation…

Ranges of the real-valued parameters $\alpha$, $a$, $b$, and $m$ are identified for which the operator $$\mathcal{A}_{\alpha}(a,b)f(x):=x^\alpha\left(f''(x)+\frac{a}{x}f'(x)+\frac{b}{x^2}f(x)\right), \quad x>0,$$ generates an analytic…

If $S=<d_1,...,d_\nu>$ is a numerical semigroup, we call the ring $\C[S]=\C[t^{d_1},...,t^{d_\nu}]$ the semigroup ring of $S$. We study the ring of differential operators on $\C[S]$, and its associated graded in the filtration induced by…

Using an effective vertex method we explicitly derive the two-loop dilatation generator of ABJM theory in its SU(2)xSU(2) sector, including all non-planar corrections. Subsequently, we apply this generator to a series of finite length…

This paper addresses extensions of the complex Ornstein-Uhlenbeck semigroup to operator algebras in free probability theory. If $a_1,...,a_k$ are $\ast$-free $\mathscr{R}$-diagonal operators in a $\mathrm{II}_1$ factor, then $D_t(a_{i_1}...…

The Cauchy problem for second order linear differential equation $u''(t)+Du'(t)+Au(t)=0$ in Hilbert space $H$ with a sectorial operator $A$ and an accretive operator $D$ is studied. Sufficient conditions for exponential decay of the…

If $a$ is a densely defined sectorial form in a Hilbert space which is possibly not closable, then we associate in a natural way a holomorphic semigroup generator with $a$. This allows us to remove in several theorems of semigroup theory…

We consider differential operators on a supermanifold of dimension $1|1$. We define non-degenerate operators as those with an invertible top coefficient in the expansion in the "superderivative" $D$ (which is the square root of the shift…