Surjectivity of Euler type differential operators on spaces of smooth functions

2019 ◽  
Vol 372 (9) ◽  
pp. 6017-6086 ◽  
Author(s):  
Paweł Domański ◽  
Michael Langenbruch
Keyword(s):  
Differential Operators ◽  
Smooth Functions

scholarly journals Mollification in Strongly Lipschitz Domains with Application to Continuous and Discrete De Rham Complexes

2016 ◽  
Vol 16 (1) ◽  
pp. 51-75 ◽  
Author(s):  
Alexandre Ern ◽  
Jean-Luc Guermond
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Vector Fields ◽  
Differential Operators ◽  
Lipschitz Domains ◽  
Projection Operators ◽  
Approximation Properties ◽  
Smooth Functions ◽  
Conforming Finite Element ◽  
De Rham

AbstractWe construct mollification operators in strongly Lipschitz domains that do not invoke non-trivial extensions, are Lp stable for any real number ${p\in [1,\infty ]}$, and commute with the differential operators ∇, ${\nabla {\times }}$, and ${\nabla {\cdot }}$. We also construct mollification operators satisfying boundary conditions and use them to characterize the kernel of traces related to the tangential and normal trace of vector fields. We use the mollification operators to build projection operators onto general H1-, ${{H}(\mathrm {curl})}$- and ${{H}(\mathrm {div})}$-conforming finite element spaces, with and without homogeneous boundary conditions. These operators commute with the differential operators ∇, ${\nabla {\times }}$, and ${\nabla {\cdot }}$, are Lp-stable, and have optimal approximation properties on smooth functions.

scholarly journals Intrinsic stochastic differential equations as jets

2018 ◽  
Vol 474 (2210) ◽  
pp. 20170559 ◽  
Author(s):  
J. Armstrong ◽  
D. Brigo
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Scheme ◽  
Differential Operators ◽  
Geometric Interpretation ◽  
Graphical Representations ◽  
Smooth Functions ◽  
One Dimensional ◽  
Small Times

We explain how Itô stochastic differential equations (SDEs) on manifolds may be defined using 2-jets of smooth functions. We show how this relationship can be interpreted in terms of a convergent numerical scheme. We also show how jets can be used to derive graphical representations of Itô SDEs, and we show how jets can be used to derive the differential operators associated with SDEs in a coordinate-free manner. We relate jets to vector flows, giving a geometric interpretation of the Itô–Stratonovich transformation. We show how percentiles can be used to give an alternative coordinate-free interpretation of the coefficients of one-dimensional SDEs. We relate this to the jet approach. This allows us to interpret the coefficients of SDEs in terms of ‘fan diagrams’. In particular, the median of an SDE solution is associated with the drift of the SDE in Stratonovich form for small times.

scholarly journals On Elliptic Operator Theory

2017 ◽  
Vol 9 (6) ◽  
pp. 144
Author(s):  
Abdalla, Salih Abdalla
Keyword(s):  
Elliptic Operator ◽  
Differential Operators ◽  
Critical Role ◽  
Elliptic Operators ◽  
Smooth Functions ◽  
Elliptic Regularity ◽  
Partial Differentiation ◽  
Mathematical Problems ◽  
Mathematical Formulas ◽  
Real Characteristic

The theory talks about partial differentiation equations. For the this reason, elliptic operators take the portion of differential operators in general Laplace operators. The operators have clear definitions through the rule that the coefficients of the highest order derivatives are positives. The above aspects imply that crucial properties, which are the main symbols, are fixed. On the other hand, they can be equivalent, meaning that they have no real characteristic directions. To expound on the above, elliptic operators are representatives of the potential theory. The reason behind this is that they often appear in the field of electrostatics and continuum mechanics. Elliptic regularity shows that their solutions tend to be smooth functions, which means that it applies if the coefficients are smooth operators. Additionally, firm state resolutions to hyperbolic and parabolic equations apply in general to solve elliptic equations. Partial differentiation equations are important mathematical formulas that apply in solving mathematical problems. The theory plays a critical role in explaining functioning of the elliptic operator as many find it a mathematical phenomenon in how efficient it is in solving problems in mathematics. The article talks about theorems and lemmas in the elliptic operator.

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